{"id":235,"date":"2014-11-12T21:36:19","date_gmt":"2014-11-12T21:36:19","guid":{"rendered":"http:\/\/courses.candelalearning.com\/novabiology\/?post_type=chapter&#038;p=235"},"modified":"2018-07-12T23:48:52","modified_gmt":"2018-07-12T23:48:52","slug":"mendels-experiments-and-the-laws-of-probability","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/nemcc-biology1v2\/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 learning-objectives\">\r\n<h3>Learning Objective<\/h3>\r\nBy the end of this section you will be able to:,\r\n<ul>\r\n \t<li>Describe the reasons for the success of Mendel\u2019s experimental work<\/li>\r\n \t<li>Describe the expected outcomes of monohybrid crosses involving dominant and recessive alleles<\/li>\r\n \t<li>Apply the product and sum rules to calculate probabilities<\/li>\r\n<\/ul>\r\n<\/div>\r\n\r\n[caption id=\"attachment_1441\" align=\"alignright\" width=\"300\"]<img class=\" wp-image-1441\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/198\/2016\/11\/28183121\/Figure_12_01_01.jpg\" alt=\"Sketch of Gregor Mendel, a monk who wore reading glasses and a large cross.\" width=\"300\" height=\"358\" \/> Figure\u00a01. Johann Gregor Mendel is considered the father of genetics.[\/caption]\r\n\r\nJohann Gregor Mendel (1822\u20131884) (Figure\u00a01) 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.\u00a0 He ultimately settled on pea plants for his primary research.\u00a0 In 1865, Mendel presented his experimental results with nearly 30,000 pea plants to the local Natural History Society. He demonstrated that traits are transmitted faithfully 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>,[footnote]Johann Gregor Mendel, Versuche \u00fcber Pflanzenhybriden Verhandlungen des naturforschenden Vereines in Br\u00fcnn, Bd. IV f\u00fcr das Jahr, 1865 Abhandlungen, 3\u201347. (for English translation see http:\/\/www.mendelweb.org\/Mendel.plain.html)[\/footnote]\u00a0in the proceedings of the Natural History Society of Br\u00fcnn.\r\n\r\nMendel\u2019s work went virtually unnoticed by the scientific community. They believed that the process of inheritance involved a blending of parental traits producing an intermediate physical appearance in offspring.\u00a0 Offspring appear to be a \u201cblend\u201d of their parents\u2019 traits. \u00a0 The blending theory of inheritance asserted that the original parental traits were lost or absorbed by the blending in the offspring.\u00a0 We now know that this is not the case. Mendel was the first researcher to see it. Instead of continuous characteristics, Mendel worked with traits that were inherited in distinct classes.\u00a0 Mendel\u2019s choice allowed him to see, experimentally, that the traits were not blended or absorbed in the offspring, but rather kept their distinctness and could be passed on. In 1868, Mendel became the abbot of the monastery and exchanged his scientific pursuits for his pastoral duties.\u00a0 Mendel was never recognized for his extraordinary scientific contributions during his lifetime. It was not until 1900 that his work was rediscovered, reproduced, and revitalized by scientists on the brink of discovering the chromosomal basis of heredity.\r\n<h2>Mendel\u2019s Model System<\/h2>\r\nMendel\u2019s work was accomplished using the garden pea, <em>Pisum sativum<\/em>, to study inheritance. This species naturally self-fertilizes and the flower petals remain sealed tightly until after pollination, preventing pollination from other plants. The result is highly inbred, or \u201ctrue-breeding,\u201d pea plants. True-breeding plants always produce offspring identical to the parent. Mendel chose well with the garden pea\u00a0 Not only was it true-breeding, but it also reached maturity in one season. This allowed several generations to be evaluated over short time periods. Finally, large quantities of garden peas could be cultivated simultaneously, allowing Mendel to conclude that his results did not come about simply by chance.\r\n<h2>Mendelian Crosses<\/h2>\r\nMendel performed hybridizations, mating two true-breeding individuals with 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.\u00a0 To prevent 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.\r\n\r\nPlants used in first crosses were called P<sub>0<\/sub>, or parental generation, plants (Figure 2). Mendel collected the seeds belonging to the P<sub>0<\/sub> plants that resulted from each cross and grew them the following season. These offspring were called the F<sub>1<\/sub>, or the first filial (<em>filial <\/em>= offspring), generation.\u00a0 Mendel examined the characteristics in the F<sub>1<\/sub> generation of plants, then allowed them to self-fertilize naturally. He then collected and grew the seeds from the F<sub>1<\/sub> plants to produce the F<sub>2<\/sub>, 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 the results from the first two generationswere the most intriguing and became the basis for his research.\r\n\r\n[caption id=\"attachment_1442\" align=\"aligncenter\" width=\"518\"]<img class=\"size-large wp-image-1442\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/198\/2016\/11\/28183158\/Figure_12_01_02-518x1024.jpg\" 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 P generation resulted in an F_{1} generation with all violet flowers. Self-fertilization of the F_{1} generation resulted in an F_{2} generation that consisted of 705 plants with violet flowers, and 224 plants with white flowers. \" width=\"518\" height=\"1024\" \/> Figure\u00a02. 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 F<sub>1<\/sub> generation all had violet flowers. In the F<sub>2<\/sub> generation, approximately three quarters of the plants had violet flowers, and one quarter had white flowers.[\/caption]\r\n<h2>Garden Pea Characteristics Revealed the Basics of Heredity<\/h2>\r\nIn his 1865 publication, Mendel reported the results of his crosses involving seven different characteristics. 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, 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 19,959 F<sub>2<\/sub> plants alone. His findings were consistent.\r\n\r\nWhat results did Mendel find for flower color? First, he confirmed that he had true-breeding plants for white and 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. Other than flower color, Mendel confirmed that the pea plants were physically identical.\r\n\r\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, Mendel found that 100 percent of the F<sub>1<\/sub> hybrid generation had violet flowers. Most scientists would have predicted the hybrid flowers to be pale violet or to have equal numbers of violet and white flowers. In other words, the contrasting parental traits were expected to blend in the offspring. Instead, Mendel\u2019s results demonstrated that the white flower trait in the F<sub>1<\/sub> generation had completely disappeared.\r\n\r\nBut, Mendel did not stop there.\u00a0 He allowed the F<sub>1<\/sub> plants to self-fertilize.\u00a0 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. 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.\u00a0 For the other six characteristics Mendel examined, the F<sub>1<\/sub> and F<sub>2<\/sub> generations behaved in the same way as flower color. One of the two traits would disappear completely 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 1).\r\n<table>\r\n<thead>\r\n<tr>\r\n<th colspan=\"5\">Table 1. The Results of Mendel\u2019s Garden Pea Hybridizations<\/th>\r\n<\/tr>\r\n<tr>\r\n<th>Characteristic<\/th>\r\n<th>Contrasting P<sub>0<\/sub> Traits<\/th>\r\n<th>F<sub>1<\/sub> Offspring Traits<\/th>\r\n<th>F<sub>2<\/sub> Offspring Traits<\/th>\r\n<th>F<sub>2<\/sub> Trait Ratios<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>Flower color<\/td>\r\n<td>Violet vs. white<\/td>\r\n<td>100 percent violet<\/td>\r\n<td>\r\n<ul>\r\n \t<li>705 violet<\/li>\r\n \t<li>224 white<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td>3.15:1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Flower position<\/td>\r\n<td>Axial vs. terminal<\/td>\r\n<td>100 percent axial<\/td>\r\n<td>\r\n<ul>\r\n \t<li>651 axial<\/li>\r\n \t<li>207 terminal<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td>3.14:1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Plant height<\/td>\r\n<td>Tall vs. dwarf<\/td>\r\n<td>100 percent tall<\/td>\r\n<td>\r\n<ul>\r\n \t<li>787 tall<\/li>\r\n \t<li>277 dwarf<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td>2.84:1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Seed texture<\/td>\r\n<td>Round vs. wrinkled<\/td>\r\n<td>100 percent round<\/td>\r\n<td>\r\n<ul>\r\n \t<li>5,474 round<\/li>\r\n \t<li>1,850 wrinkled<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td>2.96:1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Seed color<\/td>\r\n<td>Yellow vs. green<\/td>\r\n<td>100 percent yellow<\/td>\r\n<td>\r\n<ul>\r\n \t<li>6,022 yellow<\/li>\r\n \t<li>2,001 green<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td>3.01:1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Pea pod texture<\/td>\r\n<td>Inflated vs. constricted<\/td>\r\n<td>100 percent inflated<\/td>\r\n<td>\r\n<ul>\r\n \t<li>882 inflated<\/li>\r\n \t<li>299 constricted<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td>2.95:1<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Pea pod color<\/td>\r\n<td>Green vs. yellow<\/td>\r\n<td>100 percent green<\/td>\r\n<td>\r\n<ul>\r\n \t<li>428 green<\/li>\r\n \t<li>152 yellow<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td>2.82:1<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nUpon compiling his results, Mendel concluded that the characteristics could be divided into expressed(dominant) and latent(recessive) traits. Dominant traits are observable traits inherited unchanged in a hybrid cross.\u00a0 Recessive traits become latent, or disappear, in the offspring of a hybrid cross. The recessive trait does reappear in the progeny of the hybrid offspring. In Mendel's flower color cross, the dominant trait is the violet color, while white-colored flowers carry the recessive trait. Since the recessive trait reappears in the F<sub>2<\/sub> generation, the traits remained separate, not blended, in the plants of the F<sub>1<\/sub> generation. Mendel proposed that plants possessed two copies of the trait for the flower-color characteristic, with each parent giving one of its two copies to the offspring.\u00a0 Upon observing a violet flower, the dominant trait could include two dominant versions of the characteristics or one dominant and one recessive.\u00a0 If the white flower were observed, then no dominant trait would have been passed on.\r\n\r\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.\r\n<h2>Probability Basics<\/h2>\r\nProbabilities 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 possible to calculate theoretical probabilities by dividing the number of times that an event is expected to occur by the number of times that it could occur. Empirical probabilities come from observations. Theoretical probabilities 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 for a genetic event is a round seed produced by a pea plant. In his 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> plants were subsequently self-crossed, the probability of any given F<sub>2<\/sub> offspring having round seeds was now three out of four. In other words, in a large population of F<sub>2<\/sub> offspring 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.\r\n<h3>The Product Rule and Sum Rule<\/h3>\r\nMendel demonstrated that the pea-plant characteristics were transmitted as discrete units from parent to offspring.\u00a0 Mendel also determined that different characteristics, like seed color and seed texture, were transmitted independently of one another and could be considered in separate probability analyses.\u00a0 Performing a cross between a plant with green, wrinkled seeds and a plant with yellow, round seeds still produced offspring that had a 3:1 ratio of green:yellow seeds (ignoring seed texture) and a 3:1 ratio of round:wrinkled seeds (ignoring seed color). The characteristics of color and texture did not influence each other.\r\n\r\nThe product rule of probability can be applied to this phenomenon of the independent transmission of characteristics. <span style=\"text-decoration: underline\">The product rule states<\/span> t<span style=\"text-decoration: underline\">hat the probability of two independent events occurring together can be calculated by multiplying the individual probabilities of each event occurring alone.<\/span> 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 2), and each event is expected to occur with equal probability.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th colspan=\"2\">Table 2. Twelve Equally Likely Outcomes of Rolling a Die and Flipping a Penny<\/th>\r\n<\/tr>\r\n<tr>\r\n<th>Rolling Die<\/th>\r\n<th>Flipping Penny<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>D<sub>1<\/sub><\/td>\r\n<td>P<sub>H<\/sub><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>D<sub>1<\/sub><\/td>\r\n<td>P<sub>T<\/sub><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>D<sub>2<\/sub><\/td>\r\n<td>P<sub>H<\/sub><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>D<sub>2<\/sub><\/td>\r\n<td>P<sub>T<\/sub><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>D<sub>3<\/sub><\/td>\r\n<td>P<sub>H<\/sub><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>D<sub>3<\/sub><\/td>\r\n<td>P<sub>T<\/sub><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>D<sub>4<\/sub><\/td>\r\n<td>P<sub>H<\/sub><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>D<sub>4<\/sub><\/td>\r\n<td>P<sub>T<\/sub><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>D<sub>5<\/sub><\/td>\r\n<td>P<sub>H<\/sub><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>D<sub>5<\/sub><\/td>\r\n<td>P<sub>T<\/sub><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>D<sub>6<\/sub><\/td>\r\n<td>P<sub>H<\/sub><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>D<sub>6<\/sub><\/td>\r\n<td>P<sub>T<\/sub><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nOf 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>) \u00d7 (P<sub>H<\/sub>) = (1\/6) \u00d7 (1\/2) or 1\/12 (Table 2).\u00a0 The \u201cand\u201d is a signal to apply the product rule. Consider how the product rule is applied to the dihybrid cross: the probability of having both dominant traits in the F<sub>2<\/sub> progeny is the product of the probabilities of having the dominant trait for each characteristic, as shown : 3\/4\u00a0 X\u00a0 3\/4\u00a0 =\u00a0 9\/16\r\n<p style=\"text-align: center\"><\/p>\r\nThe sum rule of probability is applied when considering two mutually exclusive outcomes that can come about by more than one pathway. The sum rule states that the probability of the occurrence of one event or the other event, of two mutually exclusive events, is the sum of their individual probabilities. The \u201cor\u201d indicates that you should apply the sum rule. 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<sub>H<\/sub>) and the quarter may be tails (Q<sub>T<\/sub>), or the quarter may be heads (Q<sub>H<\/sub>) and the penny may be tails (P<sub>T<\/sub>). Either case fulfills the outcome. By the sum rule, we calculate the probability of obtaining one head and one tail as [(P<sub>H<\/sub>) \u00d7 (Q<sub>T<\/sub>)] + [(Q<sub>H<\/sub>) \u00d7 (P<sub>T<\/sub>)] = [(1\/2) \u00d7 (1\/2)] + [(1\/2) \u00d7 (1\/2)] = 1\/2 (Table 3). Notice that we used the product rule to calculate the probability of P<sub>H<\/sub> and Q<sub>T<\/sub>, and the probability of P<sub>T<\/sub> and Q<sub>H<\/sub>, before we summed them. Again, the sum rule can be applied to show the probability of having just one dominant trait in the F<sub>2<\/sub> generation of a dihybrid cross:\u00a0 3\/16\u00a0 +\u00a0 3\/4\u00a0 =\u00a0 15\/16\r\n<p style=\"text-align: center\"><\/p>\r\n\r\n<table>\r\n<thead>\r\n<tr>\r\n<th colspan=\"2\">Table 3. The Product Rule and Sum Rule<\/th>\r\n<\/tr>\r\n<tr>\r\n<th>Product Rule<\/th>\r\n<th>Sum Rule<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>For independent events A and B, the probability (P) of them both occurring (A <em>and<\/em> B) is (P<sub>A<\/sub> \u00d7 P<sub>B<\/sub>)<\/td>\r\n<td>For mutually exclusive events A and B, the probability (P) that at least one occurs (A <em>or<\/em> B) is (P<sub>A<\/sub> + P<sub>B<\/sub>)<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nUsing probability laws in practice, it is necessary to work with large sample sizes.\u00a0 Small sample sizes are prone to deviations caused by chance. The large quantities of pea plants that Mendel examined allowed him calculate the probabilities of the traits appearing in his F<sub>2<\/sub> generation. This discovery meant that when parental traits were known, the offspring\u2019s traits could be predicted accurately even before fertilization.\r\n<h2>Section Summary<\/h2>\r\nWorking with garden pea plants, Mendel found that crosses between parents differing by one trait produced F<sub>1<\/sub> offspring that all expressed the traits of one parent. Observable traits are referred to as dominant, while non-expressed traits are described as recessive. When the offspring in Mendel\u2019s experiment were self-crossed, the F<sub>2<\/sub> offspring exhibited a 3:1 ratio, confirming that the recessive trait had been transmitted faithfully from the original P<sub>0<\/sub> parent.\u00a0 Further crosses generated identical F<sub>1<\/sub> and F<sub>2<\/sub> offspring ratios. By examining sample sizes, Mendel showed that his crosses behaved according to the laws of probability, and that the traits were inherited as independent events.\r\n\r\nTwo rules in probability can be used to find the expected proportions of offspring of different traits from different crosses. To find the probability of two or more independent events occurring together, apply the product rule and multiply the probabilities of the individual events. To find the probability of two or more events occurring in combination, apply the sum rule and add their individual probabilities together.\r\n\r\nhttps:\/\/www.openassessments.org\/assessments\/480\r\n<div class=\"textbox exercises\">\r\n<h3>Additional Self Check Questions<\/h3>\r\n<ol>\r\n \t<li>Describe one of the reasons why the garden pea was an excellent choice for studying inheritance.<\/li>\r\n \t<li>How would you perform a reciprocal cross for the characteristic of stem height in the garden pea?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Answers<\/h3>\r\n<ol>\r\n \t<li>The garden pea matured in one season and has flowers that close tightly during self-pollination. It also cultivated large quantities simultaneously.<\/li>\r\n \t<li>Two sets of P<sub>0<\/sub> parents would be used. In the first cross, pollen would be transferred from a true-breeding tall plant to the stigma of a true-breeding dwarf plant. In the second cross, pollen would be transferred from a true-breeding dwarf plant to the stigma of a true-breeding tall plant. For each cross, F<sub>1<\/sub> and F<sub>2<\/sub> offspring would be analyzed to determine if offspring traits were affected according to which parent donated each trait.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>GlossarY<\/h3>\r\n&nbsp;\r\n\r\n<strong>dominant: <\/strong>trait which showsthe same physical appearance whether an individual has two copies of the trait or one copy of the dominant trait and one copy of the recessive trait\r\n\r\n<strong>F<sub>1: <\/sub><\/strong>first filial generation in a cross; the offspring of the parental generation\r\n\r\n<strong>F<sub>2: <\/sub><\/strong>second filial generation produced when F<sub>1<\/sub> individuals are self-crossed or fertilized with each other\r\n\r\n<strong>hybridization: <\/strong>process of mating two individuals that differ with the goal of achieving a certain characteristic in their offspring\r\n\r\n<strong>P<sub>0: <\/sub><\/strong>parental generation in a cross\r\n\r\n<strong>product rule: <\/strong>probability of two independent events occurring simultaneously can be calculated by multiplying the individual probabilities of each event occurring alone\r\n\r\n<strong>recessive: <\/strong>trait that appears \u201clatent\u201d or non-expressed when the individual also carries a dominant trait for that same characteristic; when present as two identical copies, the recessive trait is expressed\r\n\r\n<strong>reciprocal cross: <\/strong>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\r\n\r\n<strong>sum rule: <\/strong>probability of the occurrence of at least one of two mutually exclusive events is the sum of their individual probabilities\r\n\r\n<strong>trait: <\/strong>variation in the physical appearance of a heritable characteristic\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objective<\/h3>\n<p>By the end of this section you will be able to:,<\/p>\n<ul>\n<li>Describe the 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 product and sum rules to calculate probabilities<\/li>\n<\/ul>\n<\/div>\n<div id=\"attachment_1441\" style=\"width: 310px\" class=\"wp-caption alignright\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1441\" class=\"wp-image-1441\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/198\/2016\/11\/28183121\/Figure_12_01_01.jpg\" alt=\"Sketch of Gregor Mendel, a monk who wore reading glasses and a large cross.\" width=\"300\" height=\"358\" \/><\/p>\n<p id=\"caption-attachment-1441\" class=\"wp-caption-text\">Figure\u00a01. Johann Gregor Mendel is considered the father of genetics.<\/p>\n<\/div>\n<p>Johann Gregor Mendel (1822\u20131884) (Figure\u00a01) 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.\u00a0 He ultimately settled on pea plants for his primary research.\u00a0 In 1865, Mendel presented his experimental results with nearly 30,000 pea plants to the local Natural History Society. He demonstrated that traits are transmitted faithfully 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>,<a class=\"footnote\" title=\"Johann Gregor Mendel, Versuche \u00fcber Pflanzenhybriden Verhandlungen des naturforschenden Vereines in Br\u00fcnn, Bd. IV f\u00fcr das Jahr, 1865 Abhandlungen, 3\u201347. (for English translation see http:\/\/www.mendelweb.org\/Mendel.plain.html)\" id=\"return-footnote-235-1\" href=\"#footnote-235-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a>\u00a0in the proceedings of the Natural History Society of Br\u00fcnn.<\/p>\n<p>Mendel\u2019s work went virtually unnoticed by the scientific community. They believed that the process of inheritance involved a blending of parental traits producing an intermediate physical appearance in offspring.\u00a0 Offspring appear to be a \u201cblend\u201d of their parents\u2019 traits. \u00a0 The blending theory of inheritance asserted that the original parental traits were lost or absorbed by the blending in the offspring.\u00a0 We now know that this is not the case. Mendel was the first researcher to see it. Instead of continuous characteristics, Mendel worked with traits that were inherited in distinct classes.\u00a0 Mendel\u2019s choice allowed him to see, experimentally, that the traits were not blended or absorbed in the offspring, but rather kept their distinctness and could be passed on. In 1868, Mendel became the abbot of the monastery and exchanged his scientific pursuits for his pastoral duties.\u00a0 Mendel was never recognized for his extraordinary scientific contributions during his lifetime. It was not until 1900 that his work was rediscovered, reproduced, and revitalized by scientists on the brink of discovering the chromosomal basis of heredity.<\/p>\n<h2>Mendel\u2019s Model System<\/h2>\n<p>Mendel\u2019s work was accomplished using the garden pea, <em>Pisum sativum<\/em>, to study inheritance. This species naturally self-fertilizes and the flower petals remain sealed tightly until after pollination, preventing pollination from other plants. The result is highly inbred, or \u201ctrue-breeding,\u201d pea plants. True-breeding plants always produce offspring identical to the parent. Mendel chose well with the garden pea\u00a0 Not only was it true-breeding, but it also reached maturity in one season. This allowed several generations to be evaluated over short time periods. Finally, large quantities of garden peas could be cultivated simultaneously, allowing Mendel to conclude that his results did not come about simply by chance.<\/p>\n<h2>Mendelian Crosses<\/h2>\n<p>Mendel performed hybridizations, mating two true-breeding individuals with 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.\u00a0 To prevent 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 crosses were called P<sub>0<\/sub>, or parental generation, plants (Figure 2). Mendel collected the seeds belonging to the P<sub>0<\/sub> plants that resulted from each cross and grew them the following season. These offspring were called the F<sub>1<\/sub>, or the first filial (<em>filial <\/em>= offspring), generation.\u00a0 Mendel examined the characteristics in the F<sub>1<\/sub> generation of plants, then allowed them to self-fertilize naturally. He then collected and grew the seeds from the F<sub>1<\/sub> plants to produce the F<sub>2<\/sub>, 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 the results from the first two generationswere the most intriguing and became the basis for his research.<\/p>\n<div id=\"attachment_1442\" style=\"width: 528px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-1442\" class=\"size-large wp-image-1442\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/198\/2016\/11\/28183158\/Figure_12_01_02-518x1024.jpg\" 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 P generation resulted in an F_{1} generation with all violet flowers. Self-fertilization of the F_{1} generation resulted in an F_{2} generation that consisted of 705 plants with violet flowers, and 224 plants with white flowers.\" width=\"518\" height=\"1024\" \/><\/p>\n<p id=\"caption-attachment-1442\" class=\"wp-caption-text\">Figure\u00a02. 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 F<sub>1<\/sub> generation all had violet flowers. In the F<sub>2<\/sub> generation, approximately three quarters of the plants had violet flowers, and one quarter had white flowers.<\/p>\n<\/div>\n<h2>Garden Pea Characteristics Revealed the Basics of Heredity<\/h2>\n<p>In his 1865 publication, Mendel reported the results of his crosses involving seven different characteristics. 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, 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 19,959 F<sub>2<\/sub> plants alone. His findings were consistent.<\/p>\n<p>What results did Mendel find for flower color? First, he confirmed that he had true-breeding plants for white and 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. Other than flower color, Mendel confirmed that 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, Mendel found that 100 percent of the F<sub>1<\/sub> hybrid generation had violet flowers. Most scientists would have predicted the hybrid flowers to be pale violet or to have equal numbers of violet and white flowers. In other words, the contrasting parental traits were expected to blend in the offspring. Instead, Mendel\u2019s results demonstrated that the white flower trait in the F<sub>1<\/sub> generation had completely disappeared.<\/p>\n<p>But, Mendel did not stop there.\u00a0 He allowed the F<sub>1<\/sub> plants to self-fertilize.\u00a0 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. 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.\u00a0 For the other six characteristics Mendel examined, the F<sub>1<\/sub> and F<sub>2<\/sub> generations behaved in the same way as flower color. One of the two traits would disappear completely 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 1).<\/p>\n<table>\n<thead>\n<tr>\n<th colspan=\"5\">Table 1. The Results of Mendel\u2019s Garden Pea Hybridizations<\/th>\n<\/tr>\n<tr>\n<th>Characteristic<\/th>\n<th>Contrasting P<sub>0<\/sub> Traits<\/th>\n<th>F<sub>1<\/sub> Offspring Traits<\/th>\n<th>F<sub>2<\/sub> Offspring Traits<\/th>\n<th>F<sub>2<\/sub> Trait Ratios<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Flower color<\/td>\n<td>Violet vs. white<\/td>\n<td>100 percent violet<\/td>\n<td>\n<ul>\n<li>705 violet<\/li>\n<li>224 white<\/li>\n<\/ul>\n<\/td>\n<td>3.15:1<\/td>\n<\/tr>\n<tr>\n<td>Flower position<\/td>\n<td>Axial vs. terminal<\/td>\n<td>100 percent axial<\/td>\n<td>\n<ul>\n<li>651 axial<\/li>\n<li>207 terminal<\/li>\n<\/ul>\n<\/td>\n<td>3.14:1<\/td>\n<\/tr>\n<tr>\n<td>Plant height<\/td>\n<td>Tall vs. dwarf<\/td>\n<td>100 percent tall<\/td>\n<td>\n<ul>\n<li>787 tall<\/li>\n<li>277 dwarf<\/li>\n<\/ul>\n<\/td>\n<td>2.84:1<\/td>\n<\/tr>\n<tr>\n<td>Seed texture<\/td>\n<td>Round vs. wrinkled<\/td>\n<td>100 percent round<\/td>\n<td>\n<ul>\n<li>5,474 round<\/li>\n<li>1,850 wrinkled<\/li>\n<\/ul>\n<\/td>\n<td>2.96:1<\/td>\n<\/tr>\n<tr>\n<td>Seed color<\/td>\n<td>Yellow vs. green<\/td>\n<td>100 percent yellow<\/td>\n<td>\n<ul>\n<li>6,022 yellow<\/li>\n<li>2,001 green<\/li>\n<\/ul>\n<\/td>\n<td>3.01:1<\/td>\n<\/tr>\n<tr>\n<td>Pea pod texture<\/td>\n<td>Inflated vs. constricted<\/td>\n<td>100 percent inflated<\/td>\n<td>\n<ul>\n<li>882 inflated<\/li>\n<li>299 constricted<\/li>\n<\/ul>\n<\/td>\n<td>2.95:1<\/td>\n<\/tr>\n<tr>\n<td>Pea pod color<\/td>\n<td>Green vs. yellow<\/td>\n<td>100 percent green<\/td>\n<td>\n<ul>\n<li>428 green<\/li>\n<li>152 yellow<\/li>\n<\/ul>\n<\/td>\n<td>2.82:1<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Upon compiling his results, Mendel concluded that the characteristics could be divided into expressed(dominant) and latent(recessive) traits. Dominant traits are observable traits inherited unchanged in a hybrid cross.\u00a0 Recessive traits become latent, or disappear, in the offspring of a hybrid cross. The recessive trait does reappear in the progeny of the hybrid offspring. In Mendel&#8217;s flower color cross, the dominant trait is the violet color, while white-colored flowers carry the recessive trait. Since the recessive trait reappears in the F<sub>2<\/sub> generation, the traits remained separate, not blended, in the plants of the F<sub>1<\/sub> generation. Mendel proposed that plants possessed two copies of the trait for the flower-color characteristic, with each parent giving one of its two copies to the offspring.\u00a0 Upon observing a violet flower, the dominant trait could include two dominant versions of the characteristics or one dominant and one recessive.\u00a0 If the white flower were observed, then no dominant trait would have been passed on.<\/p>\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<h2>Probability Basics<\/h2>\n<p>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 possible to calculate theoretical probabilities by dividing the number of times that an event is expected to occur by the number of times that it could occur. Empirical probabilities come from observations. Theoretical probabilities 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 for a genetic event is a round seed produced by a pea plant. In his 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> plants were subsequently self-crossed, the probability of any given F<sub>2<\/sub> offspring having round seeds was now three out of four. In other words, in a large population of F<sub>2<\/sub> offspring 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<h3>The Product Rule and Sum Rule<\/h3>\n<p>Mendel demonstrated that the pea-plant characteristics were transmitted as discrete units from parent to offspring.\u00a0 Mendel also determined that different characteristics, like seed color and seed texture, were transmitted independently of one another and could be considered in separate probability analyses.\u00a0 Performing a cross between a plant with green, wrinkled seeds and a plant with yellow, round seeds still produced offspring that had a 3:1 ratio of green:yellow seeds (ignoring seed texture) and a 3:1 ratio of round:wrinkled seeds (ignoring seed color). The characteristics of color and texture did not influence each other.<\/p>\n<p>The product rule of probability can be applied to this phenomenon of the independent transmission of characteristics. <span style=\"text-decoration: underline\">The product rule states<\/span> t<span style=\"text-decoration: underline\">hat the probability of two independent events occurring together can be calculated by multiplying the individual probabilities of each event occurring alone.<\/span> 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 2), and each event is expected to occur with equal probability.<\/p>\n<table>\n<thead>\n<tr>\n<th colspan=\"2\">Table 2. Twelve Equally Likely Outcomes of Rolling a Die and Flipping a Penny<\/th>\n<\/tr>\n<tr>\n<th>Rolling Die<\/th>\n<th>Flipping Penny<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>D<sub>1<\/sub><\/td>\n<td>P<sub>H<\/sub><\/td>\n<\/tr>\n<tr>\n<td>D<sub>1<\/sub><\/td>\n<td>P<sub>T<\/sub><\/td>\n<\/tr>\n<tr>\n<td>D<sub>2<\/sub><\/td>\n<td>P<sub>H<\/sub><\/td>\n<\/tr>\n<tr>\n<td>D<sub>2<\/sub><\/td>\n<td>P<sub>T<\/sub><\/td>\n<\/tr>\n<tr>\n<td>D<sub>3<\/sub><\/td>\n<td>P<sub>H<\/sub><\/td>\n<\/tr>\n<tr>\n<td>D<sub>3<\/sub><\/td>\n<td>P<sub>T<\/sub><\/td>\n<\/tr>\n<tr>\n<td>D<sub>4<\/sub><\/td>\n<td>P<sub>H<\/sub><\/td>\n<\/tr>\n<tr>\n<td>D<sub>4<\/sub><\/td>\n<td>P<sub>T<\/sub><\/td>\n<\/tr>\n<tr>\n<td>D<sub>5<\/sub><\/td>\n<td>P<sub>H<\/sub><\/td>\n<\/tr>\n<tr>\n<td>D<sub>5<\/sub><\/td>\n<td>P<sub>T<\/sub><\/td>\n<\/tr>\n<tr>\n<td>D<sub>6<\/sub><\/td>\n<td>P<sub>H<\/sub><\/td>\n<\/tr>\n<tr>\n<td>D<sub>6<\/sub><\/td>\n<td>P<sub>T<\/sub><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>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>) \u00d7 (P<sub>H<\/sub>) = (1\/6) \u00d7 (1\/2) or 1\/12 (Table 2).\u00a0 The \u201cand\u201d is a signal to apply the product rule. Consider how the product rule is applied to the dihybrid cross: the probability of having both dominant traits in the F<sub>2<\/sub> progeny is the product of the probabilities of having the dominant trait for each characteristic, as shown : 3\/4\u00a0 X\u00a0 3\/4\u00a0 =\u00a0 9\/16<\/p>\n<p style=\"text-align: center\">\n<p>The sum rule of probability is applied when considering two mutually exclusive outcomes that can come about by more than one pathway. The sum rule states that the probability of the occurrence of one event or the other event, of two mutually exclusive events, is the sum of their individual probabilities. The \u201cor\u201d indicates that you should apply the sum rule. 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<sub>H<\/sub>) and the quarter may be tails (Q<sub>T<\/sub>), or the quarter may be heads (Q<sub>H<\/sub>) and the penny may be tails (P<sub>T<\/sub>). Either case fulfills the outcome. By the sum rule, we calculate the probability of obtaining one head and one tail as [(P<sub>H<\/sub>) \u00d7 (Q<sub>T<\/sub>)] + [(Q<sub>H<\/sub>) \u00d7 (P<sub>T<\/sub>)] = [(1\/2) \u00d7 (1\/2)] + [(1\/2) \u00d7 (1\/2)] = 1\/2 (Table 3). Notice that we used the product rule to calculate the probability of P<sub>H<\/sub> and Q<sub>T<\/sub>, and the probability of P<sub>T<\/sub> and Q<sub>H<\/sub>, before we summed them. Again, the sum rule can be applied to show the probability of having just one dominant trait in the F<sub>2<\/sub> generation of a dihybrid cross:\u00a0 3\/16\u00a0 +\u00a0 3\/4\u00a0 =\u00a0 15\/16<\/p>\n<p style=\"text-align: center\">\n<table>\n<thead>\n<tr>\n<th colspan=\"2\">Table 3. The Product Rule and Sum Rule<\/th>\n<\/tr>\n<tr>\n<th>Product Rule<\/th>\n<th>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 <em>and<\/em> B) is (P<sub>A<\/sub> \u00d7 P<sub>B<\/sub>)<\/td>\n<td>For mutually exclusive events A and B, the probability (P) that at least one occurs (A <em>or<\/em> B) is (P<sub>A<\/sub> + P<sub>B<\/sub>)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Using probability laws in practice, it is necessary to work with large sample sizes.\u00a0 Small sample sizes are prone to deviations caused by chance. The large quantities of pea plants that Mendel examined allowed him calculate the probabilities of the traits appearing in his F<sub>2<\/sub> generation. This discovery meant that when parental traits were known, the offspring\u2019s traits could be predicted accurately even before fertilization.<\/p>\n<h2>Section Summary<\/h2>\n<p>Working with garden pea plants, Mendel found that crosses between parents differing by one trait produced F<sub>1<\/sub> offspring that all expressed the traits of one parent. Observable traits are referred to as dominant, while non-expressed traits are described as recessive. When the offspring in Mendel\u2019s experiment were self-crossed, the F<sub>2<\/sub> offspring exhibited a 3:1 ratio, confirming that the recessive trait had been transmitted faithfully from the original P<sub>0<\/sub> parent.\u00a0 Further crosses generated identical F<sub>1<\/sub> and F<sub>2<\/sub> offspring ratios. By examining sample sizes, Mendel showed that his crosses behaved according to the laws of probability, and that the traits were inherited as independent events.<\/p>\n<p>Two rules in probability can be used to find the expected proportions of offspring of different traits from different crosses. To find the probability of two or more independent events occurring together, apply the product rule and multiply the probabilities of the individual events. To find the probability of two or more events occurring in combination, apply the sum rule and add their individual probabilities together.<\/p>\n<p><iframe src=\"https:\/\/lumenoea.herokuapp.com\/assessments\/load?src_url=https:\/\/lumenoea.herokuapp.com\/api\/assessments\/480.xml&#38;results_end_point=https:\/\/lumenoea.herokuapp.com\/api&#38;assessment_id=480&#38;confidence_levels=true&#38;enable_start=true&#38;eid=https:\/\/courses.lumenlearning.com\/nemcc-biology1v2\/chapter\/mendels-experiments-and-the-laws-of-probability\/\" frameborder=\"0\" style=\"border:none;width:100%;height:100%;min-height:400px;\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Additional Self Check Questions<\/h3>\n<ol>\n<li>Describe one of the reasons why the garden pea was an excellent choice for studying inheritance.<\/li>\n<li>How would you perform a reciprocal cross for the characteristic of stem height in the garden pea?<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Answers<\/h3>\n<ol>\n<li>The garden pea matured in one season and has flowers that close tightly during self-pollination. It also cultivated large quantities simultaneously.<\/li>\n<li>Two sets of P<sub>0<\/sub> parents would be used. In the first cross, pollen would be transferred from a true-breeding tall plant to the stigma of a true-breeding dwarf plant. In the second cross, pollen would be transferred from a true-breeding dwarf plant to the stigma of a true-breeding tall plant. For each cross, F<sub>1<\/sub> and F<sub>2<\/sub> offspring would be analyzed to determine if offspring traits were affected according to which parent donated each trait.<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>GlossarY<\/h3>\n<p>&nbsp;<\/p>\n<p><strong>dominant: <\/strong>trait which showsthe same physical appearance whether an individual has two copies of the trait or one copy of the dominant trait and one copy of the recessive trait<\/p>\n<p><strong>F<sub>1: <\/sub><\/strong>first filial generation in a cross; the offspring of the parental generation<\/p>\n<p><strong>F<sub>2: <\/sub><\/strong>second filial generation produced when F<sub>1<\/sub> individuals are self-crossed or fertilized with each other<\/p>\n<p><strong>hybridization: <\/strong>process of mating two individuals that differ with the goal of achieving a certain characteristic in their offspring<\/p>\n<p><strong>P<sub>0: <\/sub><\/strong>parental generation in a cross<\/p>\n<p><strong>product rule: <\/strong>probability of two independent events occurring simultaneously can be calculated by multiplying the individual probabilities of each event occurring alone<\/p>\n<p><strong>recessive: <\/strong>trait that appears \u201clatent\u201d or non-expressed when the individual also carries a dominant trait for that same characteristic; when present as two identical copies, the recessive trait is expressed<\/p>\n<p><strong>reciprocal cross: <\/strong>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<\/p>\n<p><strong>sum rule: <\/strong>probability of the occurrence of at least one of two mutually exclusive events is the sum of their individual probabilities<\/p>\n<p><strong>trait: <\/strong>variation in the physical appearance of a heritable characteristic<\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-235\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Biology. <strong>Authored by<\/strong>: Open Stax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/185cbf87-c72e-48f5-b51e-f14f21b5eabd@9.17:1\/Biology\">http:\/\/cnx.org\/contents\/185cbf87-c72e-48f5-b51e-f14f21b5eabd@9.17:1\/Biology<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section><hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-235-1\">Johann Gregor Mendel, Versuche \u00fcber Pflanzenhybriden Verhandlungen des naturforschenden Vereines in Br\u00fcnn, Bd. IV f\u00fcr das Jahr, 1865 Abhandlungen, 3\u201347. (for English translation see http:\/\/www.mendelweb.org\/Mendel.plain.html) <a href=\"#return-footnote-235-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":18,"menu_order":4,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Biology\",\"author\":\"Open Stax\",\"organization\":\"\",\"url\":\"http:\/\/cnx.org\/contents\/185cbf87-c72e-48f5-b51e-f14f21b5eabd@9.17:1\/Biology\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-235","chapter","type-chapter","status-publish","hentry"],"part":231,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/nemcc-biology1v2\/wp-json\/pressbooks\/v2\/chapters\/235","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/nemcc-biology1v2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/nemcc-biology1v2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/nemcc-biology1v2\/wp-json\/wp\/v2\/users\/18"}],"version-history":[{"count":20,"href":"https:\/\/courses.lumenlearning.com\/nemcc-biology1v2\/wp-json\/pressbooks\/v2\/chapters\/235\/revisions"}],"predecessor-version":[{"id":1669,"href":"https:\/\/courses.lumenlearning.com\/nemcc-biology1v2\/wp-json\/pressbooks\/v2\/chapters\/235\/revisions\/1669"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/nemcc-biology1v2\/wp-json\/pressbooks\/v2\/parts\/231"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/nemcc-biology1v2\/wp-json\/pressbooks\/v2\/chapters\/235\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/nemcc-biology1v2\/wp-json\/wp\/v2\/media?parent=235"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/nemcc-biology1v2\/wp-json\/pressbooks\/v2\/chapter-type?post=235"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/nemcc-biology1v2\/wp-json\/wp\/v2\/contributor?post=235"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/nemcc-biology1v2\/wp-json\/wp\/v2\/license?post=235"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}