{"id":854,"date":"2022-04-20T20:25:56","date_gmt":"2022-04-20T20:25:56","guid":{"rendered":"https:\/\/pressbooks.hccfl.edu\/bio1\/chapter\/mendels-experiments-and-the-laws-of-probability\/"},"modified":"2025-08-29T19:10:07","modified_gmt":"2025-08-29T19:10:07","slug":"mendels-experiments-and-the-laws-of-probability","status":"publish","type":"chapter","link":"https:\/\/pressbooks.hccfl.edu\/bio1\/chapter\/mendels-experiments-and-the-laws-of-probability\/","title":{"raw":"Mendel\u2019s Experiments and the Laws of Probability","rendered":"Mendel\u2019s Experiments and the Laws of Probability"},"content":{"raw":"<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\n<h2 class=\"textbox__title\">Learning Objectives<\/h2>\n<\/header>\n<div class=\"textbox__content\">\n\nBy the end of this section, you will be able to do the following:\n<ul>\n \t<li>Describe the scientific reasons for the success of Mendel\u2019s experimental work<\/li>\n \t<li>Describe the expected outcomes of monohybrid crosses involving dominant and recessive alleles<\/li>\n \t<li>Apply the sum and product rules to calculate probabilities<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<figure data-id=\"fig-ch12_01_01\"><span id=\"fs-id1457343\" data-type=\"media\" data-alt=\"Sketch of Gregor Mendel, a monk who wore reading glasses and a large cross.\"><\/span><\/figure>\n<span class=\"os-caption\">\u00a0<\/span>\n\n[caption id=\"attachment_852\" align=\"aligncenter\" width=\"450\"]<img class=\"wp-image-852 size-full\" src=\"http:\/\/pressbooks.hccfl.edu\/bio1\/wp-content\/uploads\/sites\/106\/2022\/04\/General-Biology-I-Lecture-Lab-1657046460_Page_560_Image_0001.jpg\" alt=\"Sketch of Gregor Mendel, a monk who wore reading glasses and a large cross.\" width=\"450\" height=\"537\"> Figure\u00a012.2\u00a0Johann Gregor Mendel is considered the father of genetics.[\/caption]\n<p style=\"text-align: justify\">Johann Gregor Mendel (1822\u20131884) (Figure 12.2) was a lifelong learner, teacher, scientist, and man of faith. As a young adult, he joined the Augustinian Abbey of St. Thomas in Brno in what is now the Czech Republic. Supported by the monastery, he taught physics, botany, and natural science courses at the secondary and university levels. In 1856, he began a decade-long research pursuit involving inheritance patterns in honeybees and plants, ultimately settling on pea plants as his primary <strong>model system<\/strong>. In 1865, Mendel presented the results of his experiments with nearly 30,000 pea plants to the local Natural History Society. He demonstrated that traits are transmitted from parents to offspring independently of other traits and in dominant and recessive patterns. In 1866, he published his work <em>Experiments in Plant Hybridization<\/em><sup>1<\/sup> in the proceedings of the Natural History Society of Br\u00fcnn.<\/p>\n<p style=\"text-align: justify\">Mendel\u2019s work went virtually unnoticed by the scientific community, which believed, incorrectly, that the process of inheritance involved a blending of parental traits that produced an intermediate physical appearance in offspring. The <strong>blending theory of inheritance <\/strong>asserted\u00a0 that the original parental traits were lost or absorbed by the blending in the offspring, but we now know that this is not the case. The long-held belief was the \u201cblending\u201d hypothesis in which genetic material contributed by the two parents mixes to literally form a blend, giving rise to uniformity in a population after several mating generations, in contrast to Mendel\u2019s \u201cparticulate\u201d hypothesis of inheritance, the real gene idea of parents passing discrete heritable genes that retain their separate original identities in offspring and are randomly inherited and can appear as traits or remain discrete in generations where the traits are skipped.<\/p>\n<p style=\"text-align: justify\">Another complicating observation in <strong>genetics<\/strong> is that traits can fall in the category of discontinuous traits or continuous traits. Discontinuous traits are of the \u201cone or the other\u201d type. For example, the peas of a pea plant are either green or yellow. Another example is the color of the flower in the pea plant. It is either violet or it is white. Now, it can be more complicated if a trait is a continuous trait. Continuous traits are of the \u201crange of\u201d phenotypes. An example could be the range of height in adult humans. Some are 4 feet (ft), some are 4 feet 1 inch (in), 4 ft 2 in, 4 ft 3 in, 5ft 9 in, 6ft 4 in, and so on. Studying the underlying genetics by looking at phenotypes of continuous traits is more complicated. Mendel chose to look at the less complicated discontinuous traits and tried to understand the underlying genetics. Mendel worked with traits that were inherited as discontinuous (specifically, violet versus white flowers). Mendel\u2019s choice of these kinds of traits allowed him to see experimentally that the traits were not blended in the offspring, nor were they absorbed; rather, they kept their distinctness and could be passed on. In 1868, Mendel became abbot of the monastery and exchanged his scientific pursuits for his pastoral duties. He was not recognized for his extraordinary scientific contributions during his lifetime. In fact, it was not until 1900 that his work was rediscovered, reproduced, and revitalized by scientists on the brink of discovering the <strong>chromosomal basis of heredity<\/strong>.<\/p>\n\n<h3 data-type=\"title\">Mendel\u2019s Model System<\/h3>\n<section id=\"fs-id2574310\" data-depth=\"1\">Mendel\u2019s seminal work was accomplished using the garden pea, <em>Pisum sativum<\/em>, to study inheritance. This species naturally self-fertilizes, such that pollen encounters ova within individual flowers. The flower petals remain sealed tightly until after pollination, preventing pollination from other plants. The result is highly inbred, or \"<strong>true-breeding<\/strong>,\"\u00a0pea plants. These are plants that <strong>always produce offspring that look like the parent<\/strong>. By experimenting with true-breeding pea plants, Mendel avoided the appearance of unexpected traits in offspring that might occur if the plants were not <strong>true breeding<\/strong>. The garden pea also grows to maturity within one season, meaning that several generations could be evaluated over a relatively short time. Finally, large quantities of garden peas could be cultivated simultaneously, allowing Mendel to conclude that his results did not come about simply by chance.<\/section><section id=\"fs-id1799593\" data-depth=\"1\">\n<h3 data-type=\"title\">Mendelian Crosses<\/h3>\nMendel performed <strong>hybridizations<\/strong>, which involve mating two true-breeding individuals that have different traits. In the pea, which is naturally self-pollinating, this is done by manually transferring pollen from the anther of a mature pea plant of one variety to the stigma of a separate mature pea plant of the second variety. In plants, pollen carries the male gametes (sperm) to the stigma, a sticky organ that traps pollen and allows the sperm to move down the pistil to the female gametes (ova) below. To prevent the pea plant that was receiving pollen from self-fertilizing and confounding his results, Mendel painstakingly removed all of the anthers from the plant\u2019s flowers before they had a chance to mature.\n\nPlants used in first-generation crosses were called <strong>P<sub>0<\/sub><\/strong>, or parental generation one (Figure 12.3). After each cross, Mendel collected the seeds belonging to the P<sub>0<\/sub> plants and grew them the following season. These offspring were called the <strong>F<sub>1<\/sub><\/strong>, or the first filial (<em>filial <\/em>= offspring, daughter or son) generation. Once Mendel examined the characteristics in the F<sub>1<\/sub> generation of plants, he allowed them to self-fertilize naturally. He then collected and grew the seeds from the F<sub>1<\/sub> plants to produce the <strong>F<sub>2<\/sub><\/strong>, or second filial, generation. Mendel\u2019s experiments extended beyond the F<sub>2<\/sub> generation to the F<sub>3<\/sub> and F<sub>4<\/sub> generations, and so on, but it was the ratio of characteristics in the P<sub>0<\/sub>\u2212F<sub>1<\/sub>\u2212F<sub>2<\/sub> generations that were the most intriguing and became the basis for Mendel\u2019s postulates.\n<div id=\"fig-ch12_01_02\" class=\"os-figure\">\n<figure data-id=\"fig-ch12_01_02\"><span id=\"fs-id2019116\" data-type=\"media\" data-alt=\"The diagram shows a cross between pea plants that are true-breeding for purple flower color and plants true-breeding for white flower color. This cross-fertilization of the upper P generation resulted in an upper case F subscript 1 baseline generation with all violet flowers. Self-fertilization of the upper F subscript one baseline generation resulted in an upper F subscript 2 baseline generation that consisted of 705 plants with violet flowers, and 224 plants with white flowers.\"><\/span><\/figure>\n<div class=\"os-caption-container\">\n\n&nbsp;\n\n[caption id=\"attachment_853\" align=\"aligncenter\" width=\"518\"]<img class=\"wp-image-853 size-full\" src=\"http:\/\/pressbooks.hccfl.edu\/bio1\/wp-content\/uploads\/sites\/106\/2025\/08\/General-Biology-I-Lecture-Lab-1657046460_Page_563_Image_0001.jpg\" alt=\"Mendel crossed plants that were true-breeding for violet flower color with plants true-breeding for white flower color (the P generation). The resulting hybrids in the F1 generation all had violet flowers. In the F2 generation, approximately three quarters of the plants had violet flowers, and one quarter had white flowers.\" width=\"518\" height=\"981\"> Figure 12.3 In one of his experiments on inheritance patterns, Mendel crossed plants that were true-breeding for violet flower color with plants true-breeding for white flower color (the P generation). The resulting hybrids in the F1 generation all had violet flowers. In the F2 generation, approximately three quarters of the plants had violet flowers, and one quarter had white flowers.[\/caption]\n\n<\/div>\n<\/div>\n<\/section><section id=\"fs-id1396419\" data-depth=\"1\">\n<h3 data-type=\"title\">Garden Pea Characteristics Revealed the Basics of Heredity<\/h3>\n<div class=\"os-table os-top-titled-container\">\n<div class=\"os-table-title\">\n\nIn his 1865 publication, Mendel reported the results of his crosses involving seven different characteristics, each with two contrasting traits. A<strong> trait<\/strong> is defined as a variation in the physical appearance of a heritable characteristic. The characteristics included plant height, seed texture, seed color, flower color, pea pod size, pea pod color, and flower position. For the characteristic of flower color, for example, the two contrasting traits were white versus violet. To fully examine each characteristic, Mendel generated large numbers of F<sub>1<\/sub> and F<sub>2 <\/sub>plants, reporting results from 19,959 F<sub>2 <\/sub>plants alone. His findings were consistent. What results did Mendel find in his crosses for flower color? First, Mendel confirmed that he had plants that bred true for white or violet flower color. Regardless of how many generations Mendel examined, all self-crossed offspring of parents with white flowers had white flowers, and all self-crossed offspring of parents with violet flowers had violet flowers. In addition, Mendel confirmed that, other than flower color, the pea plants were physically identical.\n\nOnce these validations were complete, Mendel applied the pollen from a plant with violet flowers to the stigma of a plant with white flowers. After gathering and sowing the seeds that resulted from this cross, <em>Mendel found that 100 percent of the F<sub>1<\/sub> hybrid <\/em><em>generation had violet flowers<\/em>. Mendel\u2019s results demonstrated that the white flower trait in the F<sub>1<\/sub> generation had completely disappeared. Thereby, he disproved the conventional wisdom at that time of the blending theory which would have predicted the hybrid flowers to be pale violet or for hybrid plants to have equal numbers of white and violet flowers.\n\nImportantly, Mendel did not stop his experimentation there. He discovered that the white flower trait re-appeared. \u00a0He allowed the F<sub>1<\/sub> plants to self-fertilize and found that, of F<sub>2<\/sub>-generation plants, 705 had violet flowers and 224 had white flowers. This was a ratio of 3.15 violet flowers per one white flower, or approximately 3:1. The reappearance of white flowers is important, as it hinted at something called recessive traits (see further explanation later in this paragraph). When Mendel transferred pollen from a plant with violet flowers to the stigma of a plant with white flowers and vice versa, he obtained about the same ratio regardless of which parent, male or female, contributed which trait. This is called a <strong>reciprocal cross<\/strong> \u2014 a paired cross in which the respective traits of the male and female in one cross become the respective traits of the female and male in the other cross. For the other six characteristics Mendel examined, the F<sub>1<\/sub> and F<sub>2 <\/sub>generations behaved in the same way as they had for flower color. One of the two traits would disappear or <strong>recede<\/strong>\u00a0completely from the F<sub>1<\/sub> generation only to reappear in the F<sub>2 <\/sub>generation at a ratio of approximately 3:1 (Table 12.1).\n<p style=\"text-align: center\"><strong>The Results of Mendel\u2019s Garden Pea Hybridizations<\/strong><\/p>\n\n<\/div>\n<table id=\"tab-ch12-01-01\" class=\"top-titled\" style=\"width: 737px\">\n<thead>\n<tr>\n<th style=\"width: 108.4px\" scope=\"col\">Characteristic<\/th>\n<th style=\"width: 139.7px\" scope=\"col\">Contrasting P<sub>0<\/sub>\u00a0Traits<\/th>\n<th style=\"width: 127.275px\" scope=\"col\">F<sub>1<\/sub>\u00a0Offspring Traits<\/th>\n<th style=\"width: 130.887px\" scope=\"col\">F<sub>2<\/sub>\u00a0Offspring Traits<\/th>\n<th style=\"width: 159.988px\" scope=\"col\">F<sub>2<\/sub>\u00a0Trait Ratios<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"width: 108.9px\">Flower color<\/td>\n<td style=\"width: 140.7px\">Violet vs. white<\/td>\n<td style=\"width: 128.275px\">100 percent violet<\/td>\n<td style=\"width: 131.887px\">\n<ul id=\"eip-547\" data-bullet-style=\"none\">\n \t<li>705 violet<\/li>\n \t<li>224 white<\/li>\n<\/ul>\n<\/td>\n<td style=\"width: 160.488px\">3.15:1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 108.9px\">Flower position<\/td>\n<td style=\"width: 140.7px\">Axial vs. terminal<\/td>\n<td style=\"width: 128.275px\">100 percent axial<\/td>\n<td style=\"width: 131.887px\">\n<ul id=\"fs-id3032232\" data-bullet-style=\"none\">\n \t<li>651 axial<\/li>\n \t<li>207 terminal<\/li>\n<\/ul>\n<\/td>\n<td style=\"width: 160.488px\">3.14:1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 108.9px\">Plant height<\/td>\n<td style=\"width: 140.7px\">Tall vs. dwarf<\/td>\n<td style=\"width: 128.275px\">100 percent tall<\/td>\n<td style=\"width: 131.887px\">\n<ul id=\"eip-189\" data-bullet-style=\"none\">\n \t<li>787 tall<\/li>\n \t<li>277 dwarf<\/li>\n<\/ul>\n<\/td>\n<td style=\"width: 160.488px\">2.84:1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 108.9px\">Seed texture<\/td>\n<td style=\"width: 140.7px\">Round vs. wrinkled<\/td>\n<td style=\"width: 128.275px\">100 percent round<\/td>\n<td style=\"width: 131.887px\">\n<ul id=\"fs-id1684663\" data-bullet-style=\"none\">\n \t<li>5,474 round<\/li>\n \t<li>1,850 wrinkled<\/li>\n<\/ul>\n<\/td>\n<td style=\"width: 160.488px\">2.96:1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 108.9px\">Seed color<\/td>\n<td style=\"width: 140.7px\">Yellow vs. green<\/td>\n<td style=\"width: 128.275px\">100 percent yellow<\/td>\n<td style=\"width: 131.887px\">\n<ul id=\"fs-id1808010\" data-bullet-style=\"none\">\n \t<li>6,022 yellow<\/li>\n \t<li>2,001 green<\/li>\n<\/ul>\n<\/td>\n<td style=\"width: 160.488px\">3.01:1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 108.9px\">Pea pod texture<\/td>\n<td style=\"width: 140.7px\">Inflated vs. constricted<\/td>\n<td style=\"width: 128.275px\">100 percent inflated<\/td>\n<td style=\"width: 131.887px\">\n<ul id=\"fs-id2026434\" data-bullet-style=\"none\">\n \t<li>882 inflated<\/li>\n \t<li>299 constricted<\/li>\n<\/ul>\n<\/td>\n<td style=\"width: 160.488px\">2.95:1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 108.9px\">Pea pod color<\/td>\n<td style=\"width: 140.7px\">Green vs. yellow<\/td>\n<td style=\"width: 128.275px\">100 percent green<\/td>\n<td style=\"width: 131.887px\">\n<ul id=\"fs-id1808846\" data-bullet-style=\"none\">\n \t<li>428 green<\/li>\n \t<li>152 yellow<\/li>\n<\/ul>\n<\/td>\n<td style=\"width: 160.488px\">2.82:1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\"><strong><span class=\"os-title-label\">Table<\/span>\u00a0<span class=\"os-number\">12.1<\/span><\/strong><\/div>\n<\/div>\n<div>\n\nUpon compiling his results for many thousands of plants, Mendel concluded that the characteristics could be divided into expressed and latent traits. He called these, respectively, dominant and recessive traits. <strong>Dominant traits <\/strong>are those that are inherited unchanged in a hybridization<strong>.<\/strong>\u00a0<strong>Recessive traits <\/strong>become latent, or <strong>disappear<\/strong>, in the <strong>offspring of a hybridization<\/strong>. The recessive trait does, however, reappear in the progeny of the hybrid offspring. An example of a dominant trait is the violet-flower trait. For this same characteristic (flower color), white-colored flowers are a recessive trait. The fact that the recessive trait <span style=\"font-size: 1em\">reappeared in the F2 generation meant that the traits remained separate (not blended) in the plants of the F<sub>1<\/sub> generation. Mendel also proposed that plants possessed two copies of the trait for the flower-color characteristic, and that each parent transmitted one of its two copies to its offspring, where they came together. Moreover, the physical observation of a dominant trait could mean that the genetic composition of the organism included two dominant versions of the characteristic or that it included one dominant and one recessive version. Conversely, the observation of a recessive trait meant that the organism lacked any dominant versions of this characteristic.<\/span>\n\n<\/div>\nSo why did Mendel repeatedly obtain 3:1 ratios in his crosses? To understand how Mendel deduced the basic mechanisms of inheritance that lead to such ratios, we must first review the laws of probability.\n\n<\/section><section id=\"fs-id2003118\" data-depth=\"1\">\n<h3 data-type=\"title\"><strong>Probability Basics<\/strong><\/h3>\n<p id=\"fs-id1771773\">Probabilities are mathematical measures of likelihood. The empirical probability of an event is calculated by dividing the number of times the event occurs by the total number of opportunities for the event to occur. It is also possible to calculate theoretical probabilities by dividing the number of times that an event is\u00a0<em data-effect=\"italics\">expected<\/em>\u00a0to occur by the number of times that it could occur. Empirical probabilities come from observations, like those of Mendel. Theoretical probabilities, on the other hand, come from knowing how the events are produced and assuming that the probabilities of individual outcomes are equal. A probability of one for some event indicates that it is guaranteed to occur, whereas a probability of zero indicates that it is guaranteed not to occur. An example of a genetic event is a round seed produced by a pea plant.<\/p>\n<p id=\"fs-id1771774\">In one experiment, Mendel demonstrated that the probability of the event \u201cround seed\u201d occurring was one in the F<sub>1<\/sub> offspring of true-breeding parents, one of which has round seeds and one of which has wrinkled seeds. When the F<sub>1<\/sub>\u00a0plants were subsequently self-crossed, the probability of any given F<sub>2<\/sub>\u00a0offspring having round seeds was now three out of four. In other words, in a large population of F<sub>2<\/sub>\u00a0offspring chosen at random, 75 percent were expected to have round seeds, whereas 25 percent were expected to have wrinkled seeds. Using large numbers of crosses, Mendel was able to calculate probabilities and use these to predict the outcomes of other crosses.<\/p>\n\n<section id=\"fs-id1805576\" data-depth=\"2\">\n<h4 data-type=\"title\">The Product Rule and Sum Rule<\/h4>\n<p id=\"fs-id1780876\">The\u00a0<strong><span id=\"term484\" data-type=\"term\">product rule<\/span><\/strong>\u00a0of probability can be applied to this phenomenon of the independent transmission of characteristics. The product rule states that the probability of two independent events occurring together can be calculated by multiplying the individual probabilities of each event occurring alone. To demonstrate the product rule, imagine that you are rolling a six-sided die (D) and flipping a penny (P) at the same time. The die may roll any number from 1\u20136 (D<sub>#<\/sub>), whereas the penny may turn up heads (P<sub>H<\/sub>) or tails (P<sub>T<\/sub>). The outcome of rolling the die has no effect on the outcome of flipping the penny and vice versa. There are 12 possible outcomes of this action (Table 12.2), and each event is expected to occur with equal probability.<\/p>\n\n<div class=\"os-table os-top-titled-container\">\n<div class=\"os-table-title\" style=\"text-align: center\"><strong>Twelve Equally Likely Outcomes of Rolling a Die and Flipping a Penny<\/strong><\/div>\n<table id=\"tab-ch12-01-02\" class=\"top-titled aligncenter\">\n<thead>\n<tr>\n<th scope=\"col\" data-align=\"center\">Rolling Die<\/th>\n<th scope=\"col\" data-align=\"center\">Flipping Penny<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td data-align=\"center\">D<sub>1<\/sub><\/td>\n<td data-align=\"center\">P<sub>H<\/sub><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">D<sub>1<\/sub><\/td>\n<td data-align=\"center\">P<sub>T<\/sub><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">D<sub>2<\/sub><\/td>\n<td data-align=\"center\">P<sub>H<\/sub><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">D<sub>2<\/sub><\/td>\n<td data-align=\"center\">P<sub>T<\/sub><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">D<sub>3<\/sub><\/td>\n<td data-align=\"center\">P<sub>H<\/sub><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">D<sub>3<\/sub><\/td>\n<td data-align=\"center\">P<sub>T<\/sub><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">D<sub>4<\/sub><\/td>\n<td data-align=\"center\">P<sub>H<\/sub><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">D<sub>4<\/sub><\/td>\n<td data-align=\"center\">P<sub>T<\/sub><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">D<sub>5<\/sub><\/td>\n<td data-align=\"center\">P<sub>H<\/sub><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">D<sub>5<\/sub><\/td>\n<td data-align=\"center\">P<sub>T<\/sub><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">D<sub>6<\/sub><\/td>\n<td data-align=\"center\">P<sub>H<\/sub><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">D<sub>6<\/sub><\/td>\n<td data-align=\"center\">P<sub>T<\/sub><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\" style=\"text-align: center\"><span class=\"os-title-label\">Table<\/span>\u00a0<span class=\"os-number\">12.2<\/span><\/div>\n<\/div>\n<p id=\"fs-id2316750\">Of the 12 possible outcomes, the die has a 2\/12 (or 1\/6) probability of rolling a two, and the penny has a 6\/12 (or 1\/2) probability of coming up heads. By the product rule, the probability that you will obtain the combined outcome 2 and heads is: (D<sub>2<\/sub>) x (P<sub>H<\/sub>) = (1\/6) x (1\/2) or 1\/12 (Table 12.3). Notice the word \u201cand\u201d in the description of the probability. The \u201cand\u201d is a signal to apply the product rule. For example, consider how the product rule is applied to the dihybrid cross: the probability of having both dominant traits (using for example the traits for seed, yellow and round) in the F<sub>2<\/sub> progeny is the product of the probabilities of having the dominant trait for each characteristic, as shown here:<\/p>\n<p style=\"text-align: center\"><span id=\"MJXc-Node-6\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-bevelled\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-7\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><span class=\"mjx-bevel\"><span class=\"mjx-char MJXc-TeX-size2-R\">\/<\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-8\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-10\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-12\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-bevelled\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-13\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><span class=\"mjx-bevel\"><span class=\"mjx-char MJXc-TeX-size2-R\">\/<\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-14\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-16\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-19\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-bevelled\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-20\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">9<\/span><\/span><\/span><span class=\"mjx-bevel\"><span class=\"mjx-char MJXc-TeX-size2-R\">\/<\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-21\" class=\"mjx-mrow\"><span id=\"MJXc-Node-22\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">16<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<span style=\"font-size: 1em\">On the other hand, the <\/span><strong><span id=\"term485\" style=\"font-size: 1em\" data-type=\"term\">sum rule<\/span><\/strong><span style=\"font-size: 1em\">\u00a0of probability is applied when considering two mutually exclusive outcomes that can come about by more than one pathway. The sum rule states that the <strong>probability of the occurrence of one event or the other event, of two mutually exclusive events, is the sum of their individual probabilities.<\/strong> Notice the word \u201cor\u201d in the description of the probability. The \u201cor\u201d indicates that you should apply the sum rule. In this case, let\u2019s imagine you are flipping a penny (P) and a quarter (Q). What is the probability of one coin coming up heads and one coin coming up tails? This outcome can be achieved by two cases: the penny may be heads (P<\/span><sub>H<\/sub><span style=\"font-size: 1em\">) and the quarter may be tails (Q<\/span><sub>T<\/sub><span style=\"font-size: 1em\">), or the quarter may be heads (Q<\/span><sub>H<\/sub><span style=\"font-size: 1em\">) and the penny may be tails (P<\/span><sub>T<\/sub><span style=\"font-size: 1em\">). Either case fulfills the outcome. By the sum rule, we calculate the probability of obtaining one head and one tail as [(P<\/span><sub>H<\/sub><span style=\"font-size: 1em\">) \u00d7 (Q<\/span><sub>T<\/sub><span style=\"font-size: 1em\">)] + [(Q<\/span><sub>H<\/sub><span style=\"font-size: 1em\">) \u00d7 (P<\/span><sub>T<\/sub><span style=\"font-size: 1em\">)] = [(1\/2) \u00d7 (1\/2)] + [(1\/2) \u00d7 (1\/2)] = 1\/2<\/span><span style=\"font-size: 1em\">. You should also notice that we used the product rule to calculate the probability of P<\/span><sub>H<\/sub><span style=\"font-size: 1em\">\u00a0and Q<\/span><sub>T<\/sub><span style=\"font-size: 1em\">, and also the probability of P<\/span><sub>T<\/sub><span style=\"font-size: 1em\">\u00a0and Q<\/span><sub>H<\/sub><span style=\"font-size: 1em\">, before we summed them. Again, the sum rule can be applied to show the probability of having at least one dominant trait in the F<\/span><sub>2<\/sub><span style=\"font-size: 1em\"> generation of a dihybrid cross:<\/span>\n<p style=\"text-align: center\"><span id=\"MJXc-Node-28\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size2-R\">(<\/span><\/span><span id=\"MJXc-Node-29\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-bevelled\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-30\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><\/span><span class=\"mjx-bevel\"><span class=\"mjx-char MJXc-TeX-size2-R\">\/<\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-31\" class=\"mjx-mrow\"><span id=\"MJXc-Node-32\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-33\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-34\" class=\"mjx-mfrac MJXc-space2\"><span class=\"mjx-box MJXc-bevelled\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-35\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><span class=\"mjx-bevel\"><span class=\"mjx-char MJXc-TeX-size2-R\">\/<\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-36\" class=\"mjx-mrow\"><span id=\"MJXc-Node-37\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-38\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size2-R\">)<\/span><\/span><span id=\"MJXc-Node-39\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">+<\/span><\/span><span id=\"MJXc-Node-40\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-size2-R\">(<\/span><\/span><span id=\"MJXc-Node-41\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-bevelled\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-42\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><span class=\"mjx-bevel\"><span class=\"mjx-char MJXc-TeX-size2-R\">\/<\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-43\" class=\"mjx-mrow\"><span id=\"MJXc-Node-44\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-45\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-46\" class=\"mjx-mfrac MJXc-space2\"><span class=\"mjx-box MJXc-bevelled\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-47\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><\/span><span class=\"mjx-bevel\"><span class=\"mjx-char MJXc-TeX-size2-R\">\/<\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-48\" class=\"mjx-mrow\"><span id=\"MJXc-Node-49\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-50\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size2-R\">)<\/span><\/span><span id=\"MJXc-Node-51\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-52\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-bevelled\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-53\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><span class=\"mjx-bevel\"><span class=\"mjx-char MJXc-TeX-size2-R\">\/<\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-54\" class=\"mjx-mrow\"><span id=\"MJXc-Node-55\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">16<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-56\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">+<\/span><\/span><span id=\"MJXc-Node-57\" class=\"mjx-mfrac MJXc-space2\"><span class=\"mjx-box MJXc-bevelled\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-58\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><span class=\"mjx-bevel\"><span class=\"mjx-char MJXc-TeX-size2-R\">\/<\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-59\" class=\"mjx-mrow\"><span id=\"MJXc-Node-60\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">16<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-61\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-62\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-bevelled\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-63\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><\/span><span class=\"mjx-bevel\"><span class=\"mjx-char MJXc-TeX-size2-R\">\/<\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-64\" class=\"mjx-mrow\"><span id=\"MJXc-Node-65\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">16<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-66\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-67\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-bevelled\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-68\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><span class=\"mjx-bevel\"><span class=\"mjx-char MJXc-TeX-size2-R\">\/<\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-69\" class=\"mjx-mrow\"><span id=\"MJXc-Node-70\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">8<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p style=\"text-align: center\">The major differences of the \u201cproduct rule\u201d and the \u201csum rule\u201d are summarized in (Table 12.3).<\/p>\n<p style=\"text-align: center\"><strong><span style=\"font-size: 1em\">The Product Rule and Sum Rule<\/span><\/strong><\/p>\n\n<div class=\"os-table os-top-titled-container\">\n<table id=\"tab-ch12-01-03\" class=\"top-titled aligncenter\">\n<thead>\n<tr>\n<th scope=\"col\">Product Rule<\/th>\n<th scope=\"col\">Sum Rule<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>For independent events A and B, the probability (P) of them both occurring (A\u00a0<em data-effect=\"italics\">and<\/em>\u00a0B) is (P<sub>A<\/sub>\u00a0\u00d7 P<sub>B<\/sub>)<\/td>\n<td>For mutually exclusive events A and B, the probability (P) that at least one occurs (A\u00a0<em data-effect=\"italics\">or<\/em>\u00a0B) is (P<sub>A<\/sub>\u00a0+ P<sub>B<\/sub>)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\" style=\"text-align: left\"><strong><span class=\"os-title-label\">Table<\/span>\u00a0<span class=\"os-number\">12.3<\/span><\/strong><\/div>\n<\/div>\n<p id=\"fs-id1570444\">To use probability laws in practice, we must work with large sample sizes because small sample sizes are prone to deviations caused by chance. The large quantities of pea plants that Mendel examined allowed him to calculate the probabilities of the traits appearing in his F<sub>2<\/sub>\u00a0generation. As you will learn, this discovery meant that when parental traits were known, the offspring\u2019s traits could be predicted accurately even before fertilization.<\/p>\n\n<\/section><\/section>&nbsp;\n\n[h5p id=\"165\"]\n<div data-type=\"footnote-refs\">\n<h3 data-type=\"footnote-refs-title\">Footnotes<\/h3>\n<ul data-list-type=\"bulleted\" data-bullet-style=\"none\">\n \t<li id=\"fs-id2026365\" data-type=\"footnote-ref\">1 <span data-type=\"footnote-ref-content\">Johann Gregor Mendel,\u00a0<em data-effect=\"italics\">Versuche \u00fcber Pflanzenhybriden Verhandlungen des naturforschenden Vereines in Br\u00fcnn, Bd. IV f\u00fcr das Jahr<\/em>, 1865 Abhandlungen, 3\u201347. [for English translation see\u00a0<a href=\"http:\/\/openstax.org\/l\/mendel_experiments\" target=\"_blank\" rel=\"noopener nofollow\">http:\/\/www.mendelweb.org\/Mendel.plain.html<\/a>]<\/span><\/li>\n<\/ul>\n<\/div>","rendered":"<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<h2 class=\"textbox__title\">Learning Objectives<\/h2>\n<\/header>\n<div class=\"textbox__content\">\n<p>By the end of this section, you will be able to do the following:<\/p>\n<ul>\n<li>Describe the scientific reasons for the success of Mendel\u2019s experimental work<\/li>\n<li>Describe the expected outcomes of monohybrid crosses involving dominant and recessive alleles<\/li>\n<li>Apply the sum and product rules to calculate probabilities<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<figure data-id=\"fig-ch12_01_01\"><span id=\"fs-id1457343\" data-type=\"media\" data-alt=\"Sketch of Gregor Mendel, a monk who wore reading glasses and a large cross.\"><\/span><\/figure>\n<p><span class=\"os-caption\">\u00a0<\/span><\/p>\n<figure id=\"attachment_852\" aria-describedby=\"caption-attachment-852\" style=\"width: 450px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-852 size-full\" src=\"http:\/\/pressbooks.hccfl.edu\/bio1\/wp-content\/uploads\/sites\/106\/2022\/04\/General-Biology-I-Lecture-Lab-1657046460_Page_560_Image_0001.jpg\" alt=\"Sketch of Gregor Mendel, a monk who wore reading glasses and a large cross.\" width=\"450\" height=\"537\" srcset=\"https:\/\/pressbooks.hccfl.edu\/bio1\/wp-content\/uploads\/sites\/106\/2022\/04\/General-Biology-I-Lecture-Lab-1657046460_Page_560_Image_0001.jpg 450w, https:\/\/pressbooks.hccfl.edu\/bio1\/wp-content\/uploads\/sites\/106\/2022\/04\/General-Biology-I-Lecture-Lab-1657046460_Page_560_Image_0001-251x300.jpg 251w, https:\/\/pressbooks.hccfl.edu\/bio1\/wp-content\/uploads\/sites\/106\/2022\/04\/General-Biology-I-Lecture-Lab-1657046460_Page_560_Image_0001-65x78.jpg 65w, https:\/\/pressbooks.hccfl.edu\/bio1\/wp-content\/uploads\/sites\/106\/2022\/04\/General-Biology-I-Lecture-Lab-1657046460_Page_560_Image_0001-225x269.jpg 225w, https:\/\/pressbooks.hccfl.edu\/bio1\/wp-content\/uploads\/sites\/106\/2022\/04\/General-Biology-I-Lecture-Lab-1657046460_Page_560_Image_0001-350x418.jpg 350w\" sizes=\"auto, (max-width: 450px) 100vw, 450px\" \/><figcaption id=\"caption-attachment-852\" class=\"wp-caption-text\">Figure\u00a012.2\u00a0Johann Gregor Mendel is considered the father of genetics.<\/figcaption><\/figure>\n<p style=\"text-align: justify\">Johann Gregor Mendel (1822\u20131884) (Figure 12.2) was a lifelong learner, teacher, scientist, and man of faith. As a young adult, he joined the Augustinian Abbey of St. Thomas in Brno in what is now the Czech Republic. Supported by the monastery, he taught physics, botany, and natural science courses at the secondary and university levels. In 1856, he began a decade-long research pursuit involving inheritance patterns in honeybees and plants, ultimately settling on pea plants as his primary <strong>model system<\/strong>. In 1865, Mendel presented the results of his experiments with nearly 30,000 pea plants to the local Natural History Society. He demonstrated that traits are transmitted from parents to offspring independently of other traits and in dominant and recessive patterns. In 1866, he published his work <em>Experiments in Plant Hybridization<\/em><sup>1<\/sup> in the proceedings of the Natural History Society of Br\u00fcnn.<\/p>\n<p style=\"text-align: justify\">Mendel\u2019s work went virtually unnoticed by the scientific community, which believed, incorrectly, that the process of inheritance involved a blending of parental traits that produced an intermediate physical appearance in offspring. The <strong>blending theory of inheritance <\/strong>asserted\u00a0 that the original parental traits were lost or absorbed by the blending in the offspring, but we now know that this is not the case. The long-held belief was the \u201cblending\u201d hypothesis in which genetic material contributed by the two parents mixes to literally form a blend, giving rise to uniformity in a population after several mating generations, in contrast to Mendel\u2019s \u201cparticulate\u201d hypothesis of inheritance, the real gene idea of parents passing discrete heritable genes that retain their separate original identities in offspring and are randomly inherited and can appear as traits or remain discrete in generations where the traits are skipped.<\/p>\n<p style=\"text-align: justify\">Another complicating observation in <strong>genetics<\/strong> is that traits can fall in the category of discontinuous traits or continuous traits. Discontinuous traits are of the \u201cone or the other\u201d type. For example, the peas of a pea plant are either green or yellow. Another example is the color of the flower in the pea plant. It is either violet or it is white. Now, it can be more complicated if a trait is a continuous trait. Continuous traits are of the \u201crange of\u201d phenotypes. An example could be the range of height in adult humans. Some are 4 feet (ft), some are 4 feet 1 inch (in), 4 ft 2 in, 4 ft 3 in, 5ft 9 in, 6ft 4 in, and so on. Studying the underlying genetics by looking at phenotypes of continuous traits is more complicated. Mendel chose to look at the less complicated discontinuous traits and tried to understand the underlying genetics. Mendel worked with traits that were inherited as discontinuous (specifically, violet versus white flowers). Mendel\u2019s choice of these kinds of traits allowed him to see experimentally that the traits were not blended in the offspring, nor were they absorbed; rather, they kept their distinctness and could be passed on. In 1868, Mendel became abbot of the monastery and exchanged his scientific pursuits for his pastoral duties. He was not recognized for his extraordinary scientific contributions during his lifetime. In fact, it was not until 1900 that his work was rediscovered, reproduced, and revitalized by scientists on the brink of discovering the <strong>chromosomal basis of heredity<\/strong>.<\/p>\n<h3 data-type=\"title\">Mendel\u2019s Model System<\/h3>\n<section id=\"fs-id2574310\" data-depth=\"1\">Mendel\u2019s seminal work was accomplished using the garden pea, <em>Pisum sativum<\/em>, to study inheritance. This species naturally self-fertilizes, such that pollen encounters ova within individual flowers. The flower petals remain sealed tightly until after pollination, preventing pollination from other plants. The result is highly inbred, or &#8220;<strong>true-breeding<\/strong>,&#8221;\u00a0pea plants. These are plants that <strong>always produce offspring that look like the parent<\/strong>. By experimenting with true-breeding pea plants, Mendel avoided the appearance of unexpected traits in offspring that might occur if the plants were not <strong>true breeding<\/strong>. The garden pea also grows to maturity within one season, meaning that several generations could be evaluated over a relatively short time. Finally, large quantities of garden peas could be cultivated simultaneously, allowing Mendel to conclude that his results did not come about simply by chance.<\/section>\n<section id=\"fs-id1799593\" data-depth=\"1\">\n<h3 data-type=\"title\">Mendelian Crosses<\/h3>\n<p>Mendel performed <strong>hybridizations<\/strong>, which involve mating two true-breeding individuals that have different traits. In the pea, which is naturally self-pollinating, this is done by manually transferring pollen from the anther of a mature pea plant of one variety to the stigma of a separate mature pea plant of the second variety. In plants, pollen carries the male gametes (sperm) to the stigma, a sticky organ that traps pollen and allows the sperm to move down the pistil to the female gametes (ova) below. To prevent the pea plant that was receiving pollen from self-fertilizing and confounding his results, Mendel painstakingly removed all of the anthers from the plant\u2019s flowers before they had a chance to mature.<\/p>\n<p>Plants used in first-generation crosses were called <strong>P<sub>0<\/sub><\/strong>, or parental generation one (Figure 12.3). After each cross, Mendel collected the seeds belonging to the P<sub>0<\/sub> plants and grew them the following season. These offspring were called the <strong>F<sub>1<\/sub><\/strong>, or the first filial (<em>filial <\/em>= offspring, daughter or son) generation. Once Mendel examined the characteristics in the F<sub>1<\/sub> generation of plants, he allowed them to self-fertilize naturally. He then collected and grew the seeds from the F<sub>1<\/sub> plants to produce the <strong>F<sub>2<\/sub><\/strong>, or second filial, generation. Mendel\u2019s experiments extended beyond the F<sub>2<\/sub> generation to the F<sub>3<\/sub> and F<sub>4<\/sub> generations, and so on, but it was the ratio of characteristics in the P<sub>0<\/sub>\u2212F<sub>1<\/sub>\u2212F<sub>2<\/sub> generations that were the most intriguing and became the basis for Mendel\u2019s postulates.<\/p>\n<div id=\"fig-ch12_01_02\" class=\"os-figure\">\n<figure data-id=\"fig-ch12_01_02\"><span id=\"fs-id2019116\" data-type=\"media\" data-alt=\"The diagram shows a cross between pea plants that are true-breeding for purple flower color and plants true-breeding for white flower color. This cross-fertilization of the upper P generation resulted in an upper case F subscript 1 baseline generation with all violet flowers. Self-fertilization of the upper F subscript one baseline generation resulted in an upper F subscript 2 baseline generation that consisted of 705 plants with violet flowers, and 224 plants with white flowers.\"><\/span><\/figure>\n<div class=\"os-caption-container\">\n<p>&nbsp;<\/p>\n<figure id=\"attachment_853\" aria-describedby=\"caption-attachment-853\" style=\"width: 518px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-853 size-full\" src=\"http:\/\/pressbooks.hccfl.edu\/bio1\/wp-content\/uploads\/sites\/106\/2025\/08\/General-Biology-I-Lecture-Lab-1657046460_Page_563_Image_0001.jpg\" alt=\"Mendel crossed plants that were true-breeding for violet flower color with plants true-breeding for white flower color (the P generation). The resulting hybrids in the F1 generation all had violet flowers. In the F2 generation, approximately three quarters of the plants had violet flowers, and one quarter had white flowers.\" width=\"518\" height=\"981\" srcset=\"https:\/\/pressbooks.hccfl.edu\/bio1\/wp-content\/uploads\/sites\/106\/2025\/08\/General-Biology-I-Lecture-Lab-1657046460_Page_563_Image_0001.jpg 518w, https:\/\/pressbooks.hccfl.edu\/bio1\/wp-content\/uploads\/sites\/106\/2025\/08\/General-Biology-I-Lecture-Lab-1657046460_Page_563_Image_0001-158x300.jpg 158w, https:\/\/pressbooks.hccfl.edu\/bio1\/wp-content\/uploads\/sites\/106\/2025\/08\/General-Biology-I-Lecture-Lab-1657046460_Page_563_Image_0001-65x123.jpg 65w, https:\/\/pressbooks.hccfl.edu\/bio1\/wp-content\/uploads\/sites\/106\/2025\/08\/General-Biology-I-Lecture-Lab-1657046460_Page_563_Image_0001-225x426.jpg 225w, https:\/\/pressbooks.hccfl.edu\/bio1\/wp-content\/uploads\/sites\/106\/2025\/08\/General-Biology-I-Lecture-Lab-1657046460_Page_563_Image_0001-350x663.jpg 350w\" sizes=\"auto, (max-width: 518px) 100vw, 518px\" \/><figcaption id=\"caption-attachment-853\" class=\"wp-caption-text\">Figure 12.3 In one of his experiments on inheritance patterns, Mendel crossed plants that were true-breeding for violet flower color with plants true-breeding for white flower color (the P generation). The resulting hybrids in the F1 generation all had violet flowers. In the F2 generation, approximately three quarters of the plants had violet flowers, and one quarter had white flowers.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1396419\" data-depth=\"1\">\n<h3 data-type=\"title\">Garden Pea Characteristics Revealed the Basics of Heredity<\/h3>\n<div class=\"os-table os-top-titled-container\">\n<div class=\"os-table-title\">\n<p>In his 1865 publication, Mendel reported the results of his crosses involving seven different characteristics, each with two contrasting traits. A<strong> trait<\/strong> is defined as a variation in the physical appearance of a heritable characteristic. The characteristics included plant height, seed texture, seed color, flower color, pea pod size, pea pod color, and flower position. For the characteristic of flower color, for example, the two contrasting traits were white versus violet. To fully examine each characteristic, Mendel generated large numbers of F<sub>1<\/sub> and F<sub>2 <\/sub>plants, reporting results from 19,959 F<sub>2 <\/sub>plants alone. His findings were consistent. What results did Mendel find in his crosses for flower color? First, Mendel confirmed that he had plants that bred true for white or violet flower color. Regardless of how many generations Mendel examined, all self-crossed offspring of parents with white flowers had white flowers, and all self-crossed offspring of parents with violet flowers had violet flowers. In addition, Mendel confirmed that, other than flower color, the pea plants were physically identical.<\/p>\n<p>Once these validations were complete, Mendel applied the pollen from a plant with violet flowers to the stigma of a plant with white flowers. After gathering and sowing the seeds that resulted from this cross, <em>Mendel found that 100 percent of the F<sub>1<\/sub> hybrid <\/em><em>generation had violet flowers<\/em>. Mendel\u2019s results demonstrated that the white flower trait in the F<sub>1<\/sub> generation had completely disappeared. Thereby, he disproved the conventional wisdom at that time of the blending theory which would have predicted the hybrid flowers to be pale violet or for hybrid plants to have equal numbers of white and violet flowers.<\/p>\n<p>Importantly, Mendel did not stop his experimentation there. He discovered that the white flower trait re-appeared. \u00a0He allowed the F<sub>1<\/sub> plants to self-fertilize and found that, of F<sub>2<\/sub>-generation plants, 705 had violet flowers and 224 had white flowers. This was a ratio of 3.15 violet flowers per one white flower, or approximately 3:1. The reappearance of white flowers is important, as it hinted at something called recessive traits (see further explanation later in this paragraph). When Mendel transferred pollen from a plant with violet flowers to the stigma of a plant with white flowers and vice versa, he obtained about the same ratio regardless of which parent, male or female, contributed which trait. This is called a <strong>reciprocal cross<\/strong> \u2014 a paired cross in which the respective traits of the male and female in one cross become the respective traits of the female and male in the other cross. For the other six characteristics Mendel examined, the F<sub>1<\/sub> and F<sub>2 <\/sub>generations behaved in the same way as they had for flower color. One of the two traits would disappear or <strong>recede<\/strong>\u00a0completely from the F<sub>1<\/sub> generation only to reappear in the F<sub>2 <\/sub>generation at a ratio of approximately 3:1 (Table 12.1).<\/p>\n<p style=\"text-align: center\"><strong>The Results of Mendel\u2019s Garden Pea Hybridizations<\/strong><\/p>\n<\/div>\n<table id=\"tab-ch12-01-01\" class=\"top-titled\" style=\"width: 737px\">\n<thead>\n<tr>\n<th style=\"width: 108.4px\" scope=\"col\">Characteristic<\/th>\n<th style=\"width: 139.7px\" scope=\"col\">Contrasting P<sub>0<\/sub>\u00a0Traits<\/th>\n<th style=\"width: 127.275px\" scope=\"col\">F<sub>1<\/sub>\u00a0Offspring Traits<\/th>\n<th style=\"width: 130.887px\" scope=\"col\">F<sub>2<\/sub>\u00a0Offspring Traits<\/th>\n<th style=\"width: 159.988px\" scope=\"col\">F<sub>2<\/sub>\u00a0Trait Ratios<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"width: 108.9px\">Flower color<\/td>\n<td style=\"width: 140.7px\">Violet vs. white<\/td>\n<td style=\"width: 128.275px\">100 percent violet<\/td>\n<td style=\"width: 131.887px\">\n<ul id=\"eip-547\" data-bullet-style=\"none\">\n<li>705 violet<\/li>\n<li>224 white<\/li>\n<\/ul>\n<\/td>\n<td style=\"width: 160.488px\">3.15:1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 108.9px\">Flower position<\/td>\n<td style=\"width: 140.7px\">Axial vs. terminal<\/td>\n<td style=\"width: 128.275px\">100 percent axial<\/td>\n<td style=\"width: 131.887px\">\n<ul id=\"fs-id3032232\" data-bullet-style=\"none\">\n<li>651 axial<\/li>\n<li>207 terminal<\/li>\n<\/ul>\n<\/td>\n<td style=\"width: 160.488px\">3.14:1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 108.9px\">Plant height<\/td>\n<td style=\"width: 140.7px\">Tall vs. dwarf<\/td>\n<td style=\"width: 128.275px\">100 percent tall<\/td>\n<td style=\"width: 131.887px\">\n<ul id=\"eip-189\" data-bullet-style=\"none\">\n<li>787 tall<\/li>\n<li>277 dwarf<\/li>\n<\/ul>\n<\/td>\n<td style=\"width: 160.488px\">2.84:1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 108.9px\">Seed texture<\/td>\n<td style=\"width: 140.7px\">Round vs. wrinkled<\/td>\n<td style=\"width: 128.275px\">100 percent round<\/td>\n<td style=\"width: 131.887px\">\n<ul id=\"fs-id1684663\" data-bullet-style=\"none\">\n<li>5,474 round<\/li>\n<li>1,850 wrinkled<\/li>\n<\/ul>\n<\/td>\n<td style=\"width: 160.488px\">2.96:1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 108.9px\">Seed color<\/td>\n<td style=\"width: 140.7px\">Yellow vs. green<\/td>\n<td style=\"width: 128.275px\">100 percent yellow<\/td>\n<td style=\"width: 131.887px\">\n<ul id=\"fs-id1808010\" data-bullet-style=\"none\">\n<li>6,022 yellow<\/li>\n<li>2,001 green<\/li>\n<\/ul>\n<\/td>\n<td style=\"width: 160.488px\">3.01:1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 108.9px\">Pea pod texture<\/td>\n<td style=\"width: 140.7px\">Inflated vs. constricted<\/td>\n<td style=\"width: 128.275px\">100 percent inflated<\/td>\n<td style=\"width: 131.887px\">\n<ul id=\"fs-id2026434\" data-bullet-style=\"none\">\n<li>882 inflated<\/li>\n<li>299 constricted<\/li>\n<\/ul>\n<\/td>\n<td style=\"width: 160.488px\">2.95:1<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 108.9px\">Pea pod color<\/td>\n<td style=\"width: 140.7px\">Green vs. yellow<\/td>\n<td style=\"width: 128.275px\">100 percent green<\/td>\n<td style=\"width: 131.887px\">\n<ul id=\"fs-id1808846\" data-bullet-style=\"none\">\n<li>428 green<\/li>\n<li>152 yellow<\/li>\n<\/ul>\n<\/td>\n<td style=\"width: 160.488px\">2.82:1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\"><strong><span class=\"os-title-label\">Table<\/span>\u00a0<span class=\"os-number\">12.1<\/span><\/strong><\/div>\n<\/div>\n<div>\n<p>Upon compiling his results for many thousands of plants, Mendel concluded that the characteristics could be divided into expressed and latent traits. He called these, respectively, dominant and recessive traits. <strong>Dominant traits <\/strong>are those that are inherited unchanged in a hybridization<strong>.<\/strong>\u00a0<strong>Recessive traits <\/strong>become latent, or <strong>disappear<\/strong>, in the <strong>offspring of a hybridization<\/strong>. The recessive trait does, however, reappear in the progeny of the hybrid offspring. An example of a dominant trait is the violet-flower trait. For this same characteristic (flower color), white-colored flowers are a recessive trait. The fact that the recessive trait <span style=\"font-size: 1em\">reappeared in the F2 generation meant that the traits remained separate (not blended) in the plants of the F<sub>1<\/sub> generation. Mendel also proposed that plants possessed two copies of the trait for the flower-color characteristic, and that each parent transmitted one of its two copies to its offspring, where they came together. Moreover, the physical observation of a dominant trait could mean that the genetic composition of the organism included two dominant versions of the characteristic or that it included one dominant and one recessive version. Conversely, the observation of a recessive trait meant that the organism lacked any dominant versions of this characteristic.<\/span><\/p>\n<\/div>\n<p>So why did Mendel repeatedly obtain 3:1 ratios in his crosses? To understand how Mendel deduced the basic mechanisms of inheritance that lead to such ratios, we must first review the laws of probability.<\/p>\n<\/section>\n<section id=\"fs-id2003118\" data-depth=\"1\">\n<h3 data-type=\"title\"><strong>Probability Basics<\/strong><\/h3>\n<p id=\"fs-id1771773\">Probabilities are mathematical measures of likelihood. The empirical probability of an event is calculated by dividing the number of times the event occurs by the total number of opportunities for the event to occur. It is also possible to calculate theoretical probabilities by dividing the number of times that an event is\u00a0<em data-effect=\"italics\">expected<\/em>\u00a0to occur by the number of times that it could occur. Empirical probabilities come from observations, like those of Mendel. Theoretical probabilities, on the other hand, come from knowing how the events are produced and assuming that the probabilities of individual outcomes are equal. A probability of one for some event indicates that it is guaranteed to occur, whereas a probability of zero indicates that it is guaranteed not to occur. An example of a genetic event is a round seed produced by a pea plant.<\/p>\n<p id=\"fs-id1771774\">In one experiment, Mendel demonstrated that the probability of the event \u201cround seed\u201d occurring was one in the F<sub>1<\/sub> offspring of true-breeding parents, one of which has round seeds and one of which has wrinkled seeds. When the F<sub>1<\/sub>\u00a0plants were subsequently self-crossed, the probability of any given F<sub>2<\/sub>\u00a0offspring having round seeds was now three out of four. In other words, in a large population of F<sub>2<\/sub>\u00a0offspring chosen at random, 75 percent were expected to have round seeds, whereas 25 percent were expected to have wrinkled seeds. Using large numbers of crosses, Mendel was able to calculate probabilities and use these to predict the outcomes of other crosses.<\/p>\n<section id=\"fs-id1805576\" data-depth=\"2\">\n<h4 data-type=\"title\">The Product Rule and Sum Rule<\/h4>\n<p id=\"fs-id1780876\">The\u00a0<strong><span id=\"term484\" data-type=\"term\">product rule<\/span><\/strong>\u00a0of probability can be applied to this phenomenon of the independent transmission of characteristics. The product rule states that the probability of two independent events occurring together can be calculated by multiplying the individual probabilities of each event occurring alone. To demonstrate the product rule, imagine that you are rolling a six-sided die (D) and flipping a penny (P) at the same time. The die may roll any number from 1\u20136 (D<sub>#<\/sub>), whereas the penny may turn up heads (P<sub>H<\/sub>) or tails (P<sub>T<\/sub>). The outcome of rolling the die has no effect on the outcome of flipping the penny and vice versa. There are 12 possible outcomes of this action (Table 12.2), and each event is expected to occur with equal probability.<\/p>\n<div class=\"os-table os-top-titled-container\">\n<div class=\"os-table-title\" style=\"text-align: center\"><strong>Twelve Equally Likely Outcomes of Rolling a Die and Flipping a Penny<\/strong><\/div>\n<table id=\"tab-ch12-01-02\" class=\"top-titled aligncenter\">\n<thead>\n<tr>\n<th scope=\"col\" data-align=\"center\">Rolling Die<\/th>\n<th scope=\"col\" data-align=\"center\">Flipping Penny<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td data-align=\"center\">D<sub>1<\/sub><\/td>\n<td data-align=\"center\">P<sub>H<\/sub><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">D<sub>1<\/sub><\/td>\n<td data-align=\"center\">P<sub>T<\/sub><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">D<sub>2<\/sub><\/td>\n<td data-align=\"center\">P<sub>H<\/sub><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">D<sub>2<\/sub><\/td>\n<td data-align=\"center\">P<sub>T<\/sub><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">D<sub>3<\/sub><\/td>\n<td data-align=\"center\">P<sub>H<\/sub><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">D<sub>3<\/sub><\/td>\n<td data-align=\"center\">P<sub>T<\/sub><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">D<sub>4<\/sub><\/td>\n<td data-align=\"center\">P<sub>H<\/sub><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">D<sub>4<\/sub><\/td>\n<td data-align=\"center\">P<sub>T<\/sub><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">D<sub>5<\/sub><\/td>\n<td data-align=\"center\">P<sub>H<\/sub><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">D<sub>5<\/sub><\/td>\n<td data-align=\"center\">P<sub>T<\/sub><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">D<sub>6<\/sub><\/td>\n<td data-align=\"center\">P<sub>H<\/sub><\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">D<sub>6<\/sub><\/td>\n<td data-align=\"center\">P<sub>T<\/sub><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\" style=\"text-align: center\"><span class=\"os-title-label\">Table<\/span>\u00a0<span class=\"os-number\">12.2<\/span><\/div>\n<\/div>\n<p id=\"fs-id2316750\">Of the 12 possible outcomes, the die has a 2\/12 (or 1\/6) probability of rolling a two, and the penny has a 6\/12 (or 1\/2) probability of coming up heads. By the product rule, the probability that you will obtain the combined outcome 2 and heads is: (D<sub>2<\/sub>) x (P<sub>H<\/sub>) = (1\/6) x (1\/2) or 1\/12 (Table 12.3). Notice the word \u201cand\u201d in the description of the probability. The \u201cand\u201d is a signal to apply the product rule. For example, consider how the product rule is applied to the dihybrid cross: the probability of having both dominant traits (using for example the traits for seed, yellow and round) in the F<sub>2<\/sub> progeny is the product of the probabilities of having the dominant trait for each characteristic, as shown here:<\/p>\n<p style=\"text-align: center\"><span id=\"MJXc-Node-6\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-bevelled\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-7\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><span class=\"mjx-bevel\"><span class=\"mjx-char MJXc-TeX-size2-R\">\/<\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-8\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-10\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-12\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-bevelled\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-13\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><span class=\"mjx-bevel\"><span class=\"mjx-char MJXc-TeX-size2-R\">\/<\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-14\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-16\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-19\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-bevelled\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-20\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">9<\/span><\/span><\/span><span class=\"mjx-bevel\"><span class=\"mjx-char MJXc-TeX-size2-R\">\/<\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-21\" class=\"mjx-mrow\"><span id=\"MJXc-Node-22\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">16<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p><span style=\"font-size: 1em\">On the other hand, the <\/span><strong><span id=\"term485\" style=\"font-size: 1em\" data-type=\"term\">sum rule<\/span><\/strong><span style=\"font-size: 1em\">\u00a0of probability is applied when considering two mutually exclusive outcomes that can come about by more than one pathway. The sum rule states that the <strong>probability of the occurrence of one event or the other event, of two mutually exclusive events, is the sum of their individual probabilities.<\/strong> Notice the word \u201cor\u201d in the description of the probability. The \u201cor\u201d indicates that you should apply the sum rule. In this case, let\u2019s imagine you are flipping a penny (P) and a quarter (Q). What is the probability of one coin coming up heads and one coin coming up tails? This outcome can be achieved by two cases: the penny may be heads (P<\/span><sub>H<\/sub><span style=\"font-size: 1em\">) and the quarter may be tails (Q<\/span><sub>T<\/sub><span style=\"font-size: 1em\">), or the quarter may be heads (Q<\/span><sub>H<\/sub><span style=\"font-size: 1em\">) and the penny may be tails (P<\/span><sub>T<\/sub><span style=\"font-size: 1em\">). Either case fulfills the outcome. By the sum rule, we calculate the probability of obtaining one head and one tail as [(P<\/span><sub>H<\/sub><span style=\"font-size: 1em\">) \u00d7 (Q<\/span><sub>T<\/sub><span style=\"font-size: 1em\">)] + [(Q<\/span><sub>H<\/sub><span style=\"font-size: 1em\">) \u00d7 (P<\/span><sub>T<\/sub><span style=\"font-size: 1em\">)] = [(1\/2) \u00d7 (1\/2)] + [(1\/2) \u00d7 (1\/2)] = 1\/2<\/span><span style=\"font-size: 1em\">. You should also notice that we used the product rule to calculate the probability of P<\/span><sub>H<\/sub><span style=\"font-size: 1em\">\u00a0and Q<\/span><sub>T<\/sub><span style=\"font-size: 1em\">, and also the probability of P<\/span><sub>T<\/sub><span style=\"font-size: 1em\">\u00a0and Q<\/span><sub>H<\/sub><span style=\"font-size: 1em\">, before we summed them. Again, the sum rule can be applied to show the probability of having at least one dominant trait in the F<\/span><sub>2<\/sub><span style=\"font-size: 1em\"> generation of a dihybrid cross:<\/span><\/p>\n<p style=\"text-align: center\"><span id=\"MJXc-Node-28\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size2-R\">(<\/span><\/span><span id=\"MJXc-Node-29\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-bevelled\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-30\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><\/span><span class=\"mjx-bevel\"><span class=\"mjx-char MJXc-TeX-size2-R\">\/<\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-31\" class=\"mjx-mrow\"><span id=\"MJXc-Node-32\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-33\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-34\" class=\"mjx-mfrac MJXc-space2\"><span class=\"mjx-box MJXc-bevelled\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-35\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><span class=\"mjx-bevel\"><span class=\"mjx-char MJXc-TeX-size2-R\">\/<\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-36\" class=\"mjx-mrow\"><span id=\"MJXc-Node-37\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-38\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size2-R\">)<\/span><\/span><span id=\"MJXc-Node-39\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">+<\/span><\/span><span id=\"MJXc-Node-40\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-size2-R\">(<\/span><\/span><span id=\"MJXc-Node-41\" class=\"mjx-mfrac\"><span class=\"mjx-box MJXc-bevelled\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-42\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><span class=\"mjx-bevel\"><span class=\"mjx-char MJXc-TeX-size2-R\">\/<\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-43\" class=\"mjx-mrow\"><span id=\"MJXc-Node-44\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-45\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">\u00d7<\/span><\/span><span id=\"MJXc-Node-46\" class=\"mjx-mfrac MJXc-space2\"><span class=\"mjx-box MJXc-bevelled\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-47\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">1<\/span><\/span><\/span><span class=\"mjx-bevel\"><span class=\"mjx-char MJXc-TeX-size2-R\">\/<\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-48\" class=\"mjx-mrow\"><span id=\"MJXc-Node-49\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">4<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-50\" class=\"mjx-mo\"><span class=\"mjx-char MJXc-TeX-size2-R\">)<\/span><\/span><span id=\"MJXc-Node-51\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-52\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-bevelled\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-53\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><span class=\"mjx-bevel\"><span class=\"mjx-char MJXc-TeX-size2-R\">\/<\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-54\" class=\"mjx-mrow\"><span id=\"MJXc-Node-55\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">16<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-56\" class=\"mjx-mo MJXc-space2\"><span class=\"mjx-char MJXc-TeX-main-R\">+<\/span><\/span><span id=\"MJXc-Node-57\" class=\"mjx-mfrac MJXc-space2\"><span class=\"mjx-box MJXc-bevelled\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-58\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><span class=\"mjx-bevel\"><span class=\"mjx-char MJXc-TeX-size2-R\">\/<\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-59\" class=\"mjx-mrow\"><span id=\"MJXc-Node-60\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">16<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-61\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-62\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-bevelled\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-63\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">6<\/span><\/span><\/span><span class=\"mjx-bevel\"><span class=\"mjx-char MJXc-TeX-size2-R\">\/<\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-64\" class=\"mjx-mrow\"><span id=\"MJXc-Node-65\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">16<\/span><\/span><\/span><\/span><\/span><\/span><span id=\"MJXc-Node-66\" class=\"mjx-mo MJXc-space3\"><span class=\"mjx-char MJXc-TeX-main-R\">=<\/span><\/span><span id=\"MJXc-Node-67\" class=\"mjx-mfrac MJXc-space3\"><span class=\"mjx-box MJXc-bevelled\"><span class=\"mjx-numerator\"><span id=\"MJXc-Node-68\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">3<\/span><\/span><\/span><span class=\"mjx-bevel\"><span class=\"mjx-char MJXc-TeX-size2-R\">\/<\/span><\/span><span class=\"mjx-denominator\"><span id=\"MJXc-Node-69\" class=\"mjx-mrow\"><span id=\"MJXc-Node-70\" class=\"mjx-mn\"><span class=\"mjx-char MJXc-TeX-main-R\">8<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n<p style=\"text-align: center\">The major differences of the \u201cproduct rule\u201d and the \u201csum rule\u201d are summarized in (Table 12.3).<\/p>\n<p style=\"text-align: center\"><strong><span style=\"font-size: 1em\">The Product Rule and Sum Rule<\/span><\/strong><\/p>\n<div class=\"os-table os-top-titled-container\">\n<table id=\"tab-ch12-01-03\" class=\"top-titled aligncenter\">\n<thead>\n<tr>\n<th scope=\"col\">Product Rule<\/th>\n<th scope=\"col\">Sum Rule<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>For independent events A and B, the probability (P) of them both occurring (A\u00a0<em data-effect=\"italics\">and<\/em>\u00a0B) is (P<sub>A<\/sub>\u00a0\u00d7 P<sub>B<\/sub>)<\/td>\n<td>For mutually exclusive events A and B, the probability (P) that at least one occurs (A\u00a0<em data-effect=\"italics\">or<\/em>\u00a0B) is (P<sub>A<\/sub>\u00a0+ P<sub>B<\/sub>)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\" style=\"text-align: left\"><strong><span class=\"os-title-label\">Table<\/span>\u00a0<span class=\"os-number\">12.3<\/span><\/strong><\/div>\n<\/div>\n<p id=\"fs-id1570444\">To use probability laws in practice, we must work with large sample sizes because small sample sizes are prone to deviations caused by chance. The large quantities of pea plants that Mendel examined allowed him to calculate the probabilities of the traits appearing in his F<sub>2<\/sub>\u00a0generation. As you will learn, this discovery meant that when parental traits were known, the offspring\u2019s traits could be predicted accurately even before fertilization.<\/p>\n<\/section>\n<\/section>\n<p>&nbsp;<\/p>\n<div id=\"h5p-165\">\n<div class=\"h5p-iframe-wrapper\"><iframe id=\"h5p-iframe-165\" class=\"h5p-iframe\" data-content-id=\"165\" style=\"height:1px\" src=\"about:blank\" frameBorder=\"0\" scrolling=\"no\" title=\"Ch. 12 Traits\"><\/iframe><\/div>\n<\/div>\n<div data-type=\"footnote-refs\">\n<h3 data-type=\"footnote-refs-title\">Footnotes<\/h3>\n<ul data-list-type=\"bulleted\" data-bullet-style=\"none\">\n<li id=\"fs-id2026365\" data-type=\"footnote-ref\">1 <span data-type=\"footnote-ref-content\">Johann Gregor Mendel,\u00a0<em data-effect=\"italics\">Versuche \u00fcber Pflanzenhybriden Verhandlungen des naturforschenden Vereines in Br\u00fcnn, Bd. IV f\u00fcr das Jahr<\/em>, 1865 Abhandlungen, 3\u201347. [for English translation see\u00a0<a href=\"http:\/\/openstax.org\/l\/mendel_experiments\" target=\"_blank\" rel=\"noopener nofollow\">http:\/\/www.mendelweb.org\/Mendel.plain.html<\/a>]<\/span><\/li>\n<\/ul>\n<\/div>\n","protected":false},"author":130,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["jung-choi","mary-ann-clark","matthew-douglas"],"pb_section_license":"cc-by"},"chapter-type":[],"contributor":[92,93,94],"license":[53],"class_list":["post-854","chapter","type-chapter","status-publish","hentry","contributor-jung-choi","contributor-mary-ann-clark","contributor-matthew-douglas","license-cc-by"],"part":847,"_links":{"self":[{"href":"https:\/\/pressbooks.hccfl.edu\/bio1\/wp-json\/pressbooks\/v2\/chapters\/854","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.hccfl.edu\/bio1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.hccfl.edu\/bio1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.hccfl.edu\/bio1\/wp-json\/wp\/v2\/users\/130"}],"version-history":[{"count":2,"href":"https:\/\/pressbooks.hccfl.edu\/bio1\/wp-json\/pressbooks\/v2\/chapters\/854\/revisions"}],"predecessor-version":[{"id":1060,"href":"https:\/\/pressbooks.hccfl.edu\/bio1\/wp-json\/pressbooks\/v2\/chapters\/854\/revisions\/1060"}],"part":[{"href":"https:\/\/pressbooks.hccfl.edu\/bio1\/wp-json\/pressbooks\/v2\/parts\/847"}],"metadata":[{"href":"https:\/\/pressbooks.hccfl.edu\/bio1\/wp-json\/pressbooks\/v2\/chapters\/854\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.hccfl.edu\/bio1\/wp-json\/wp\/v2\/media?parent=854"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.hccfl.edu\/bio1\/wp-json\/pressbooks\/v2\/chapter-type?post=854"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.hccfl.edu\/bio1\/wp-json\/wp\/v2\/contributor?post=854"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.hccfl.edu\/bio1\/wp-json\/wp\/v2\/license?post=854"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}