In this section, you will explore the following questions:
 Why was Mendel’s experimental work so successful?
 How do the sum and product rules of probability predict the outcomes of monohybrid crosses involving dominant and recessive alleles?
Connection for AP^{®} Courses
Genetics is the science of heredity. Austrian monk Gregor Mendel set the framework for genetics long before chromosomes or genes had been identified, at a time when meiosis was not well understood. Working with garden peas, Mendel found that crosses between truebreading parents (P) that differed in one trait (e.g., color: green peas versus yellow peas) produced first generation (F1) offspring that all expressed the trait of one parent (e.g., all green or all yellow). Mendel used the term dominant to refer to the trait that was observed, and recessive to denote that nonexpressed trait, or the trait that had “disappeared” in this first generation. When the F1 offspring were crossed with each other, the F2 offspring exhibited both traits in a 3:1 ratio. Other crosses (e.g., height: tall plants versus short plants) generated the same 3:1 ratio (in this example, tall to short) in the F2 offspring. By mathematically examining sample sizes, Mendel showed that genetic crosses behaved according to the laws of probability, and that the traits were inherited as independent events. In other words, Mendel used statistical methods to build his model of inheritance.
As you have likely noticed, the AP Biology course emphasizes the application of mathematics. Two rules of probability can be used to find the expected proportions of different traits in offspring from different crosses. To find the probability of two or more independent events (events where the outcome of one event has no influence on the outcome of the other event) occurring together, apply the product rule and multiply the probabilities of the individual events. To find the probability that one of two or more events occur, apply the sum rule and add their probabilities together.
The content presented in this section supports the learning objectives outlined in Big Idea 3 of the AP^{®} Biology Curriculum Framework. The AP^{®} learning objectives merge essential knowledge content with one or more of the seven science practices. These objectives provide a transparent foundation for the AP^{®} Biology course, along with inquirybased laboratory experiences, instructional activities, and AP^{®} exam questions.
Big Idea 3  Living systems store, retrieve, transmit and respond to information essential to life processes. 
Enduring Understanding 3.A  Heritable information provides for continuity of life. 
Essential Knowledge  3.A.3 The chromosomal basis of inheritance proposed by Mendel provides an understanding of the pattern of passage of genes from parent to offspring. 
Science Practice  3.1 The student can pose scientific questions. 
Learning Objective  3.13 The student is able to pose questions about ethical, social, or medical issues surrounding human genetic disorders. 
Essential Knowledge  3.A.3 The chromosomal basis of inheritance proposed by Mendel provides an understanding of the pattern of passage of genes from parent to offspring. 
Science Practice  2.2 The student can apply mathematical routines to quantities that describe natural phenomena. 
Learning Objective  3.14 The student is able to apply mathematical routines to determine Mendelian patterns of inheritance provided by data sets. 
Teacher Support
Two rules of probability are used in solving genetics problems: the rule of multiplication and the rule of addition. The probability that independent events will occur simultaneously is the product of their individual probabilities. If two dices are tossed, what is the probability of landing two ones? A die has 6 faces, and assuming the die is not loaded, each face has the same probability of outcome. The probability of obtaining the number 1 is equal to the number on the die divided by the total number of sides: $\frac{1}{6}$. The probability of rolling two ones is equal to $\frac{1}{6}\text{\hspace{0.17em}}\times \text{\hspace{0.17em}}\frac{1}{6}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{1}{36}$.
The probability that any one of a set of mutually exclusive events will occur is the sum of their individual probabilities. The probability of rolling a 1 or a 2 is equal to $\frac{1}{6}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{1}{6}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{1}{3}$ because the two outcomes are mutually exclusive. If we roll a 1, it cannot be a 2.
Tell students that Gregor Mendel was a monk who had received a solid scientific education and had excelled at mathematics. He brought this knowledge of science into his experiments with peas.
Engage students in describing what makes a good organism to study genetics. One approach is to ask the class if they would use elephants to study genetics. The disadvantages of using elephants actually highlight the advantages of using peas, corn, fruit flies, or mice for genetics studies: short life cycle, easy to maintain and handle, large number of offspring for statistical analysis, etc.
The concepts of statistics are not intuitive. Practice with dice and coins. Explain that the probability ratios are achieved with large numbers of trials.
Dominant traits are the ones expressed in a dominant/recessive situation. They do not usually repress the recessive trait. A dominant trait is not necessarily the most common trait in a population. For example, type O blood is a recessive trait, but it is the most frequent blood group in many ethnic groups. A dominant trait can be lethal. A dominant allele is not better than the recessive allele. Whether a trait is beneficial depends on the environment. Give the example of wing color in moths. Dark pigmentation is beneficial in a polluted environment where predators would not pick up the moths on dark tree barks. For example, the population peppered moths in 19th century London shifted so that their wing colors were darker to blend in with the soot of the Industrial Revolution. After pollution levels dropped, light pigmentation became more prevalent because it helped the moths to escape notice.
Johann Gregor Mendel (1822–1884) (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 decadelong research pursuit involving inheritance patterns in honeybees and plants, ultimately settling on pea plants as his primary model system (a system with convenient characteristics used to study a specific biological phenomenon to be applied to other systems). 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 faithfully from parents to offspring independently of other traits and in dominant and recessive patterns. In 1866, he published his work, Experiments in Plant Hybridization,^{1} in the proceedings of the Natural History Society of Brünn.
Mendel’s work went virtually unnoticed by the scientific community that believed, incorrectly, that the process of inheritance involved a blending of parental traits that produced an intermediate physical appearance in offspring; this hypothetical process appeared to be correct because of what we know now as continuous variation. Continuous variation results from the action of many genes to determine a characteristic like human height. Offspring appear to be a “blend” of their parents’ traits when we look at characteristics that exhibit continuous variation. The blending theory of inheritance asserted 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. Mendel was the first researcher to see it. Instead of continuous characteristics, Mendel worked with traits that were inherited in distinct classes (specifically, violet versus white flowers); this is referred to as discontinuous variation. Mendel’s choice of these kinds of traits allowed him to see experimentally that the traits were not blended in the offspring, nor were they absorbed, but rather that 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 chromosomal basis of heredity.
Mendel’s Model System
Mendel’s seminal work was accomplished using the garden pea, Pisum sativum, to study inheritance. This species naturally selffertilizes, 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 “truebreeding,” pea plants. These are plants that always produce offspring that look like the parent. By experimenting with truebreeding pea plants, Mendel avoided the appearance of unexpected traits in offspring that might occur if the plants were not true breeding. 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.
Mendelian Crosses
Mendel performed hybridizations, which involve mating two truebreeding individuals that have different traits. In the pea, which is naturally selfpollinating, 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 selffertilizing and confounding his results, Mendel painstakingly removed all of the anthers from the plant’s flowers before they had a chance to mature.
Plants used in firstgeneration crosses were called P_{0}, or parental generation one, plants (Figure 12.3). Mendel collected the seeds belonging to the P_{0} plants that resulted from each cross and grew them the following season. These offspring were called the F_{1}, or the first filial (filial = offspring, daughter or son), generation. Once Mendel examined the characteristics in the F_{1} generation of plants, he allowed them to selffertilize naturally. He then collected and grew the seeds from the F_{1} plants to produce the F_{2}, or second filial, generation. Mendel’s experiments extended beyond the F_{2} generation to the F_{3} and F_{4 }generations, and so on, but it was the ratio of characteristics in the P_{0}−F_{1}−F_{2} generations that were the most intriguing and became the basis for Mendel’s postulates.
Garden Pea Characteristics Revealed the Basics of Heredity
In his 1865 publication, Mendel reported the results of his crosses involving seven different characteristics, each with two contrasting traits. A trait 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_{1 }and F_{2} plants, reporting results from 19,959 F_{2} 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 selfcrossed offspring of parents with white flowers had white flowers, and all selfcrossed offspring of parents with violet flowers had violet flowers. In addition, Mendel confirmed that, other than flower color, the pea plants were physically identical.
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_{1} hybrid generation had violet flowers. Conventional wisdom at that time would have predicted the hybrid flowers to be pale violet or for hybrid plants to have equal numbers of white and violet flowers. In other words, the contrasting parental traits were expected to blend in the offspring. Instead, Mendel’s results demonstrated that the white flower trait in the F_{1} generation had completely disappeared.
Importantly, Mendel did not stop his experimentation there. He allowed the F_{1} plants to selffertilize and found that, of F_{2}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. This is called a reciprocal cross—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_{1} and F_{2} generations behaved in the same way as they had for flower color. One of the two traits would disappear completely from the F_{1} generation only to reappear in the F_{2} generation at a ratio of approximately 3:1 (Table 12.1).
The Results of Mendel’s Garden Pea Hybridizations  

Characteristic  Contrasting P_{0} Traits  F_{1} Offspring Traits  F_{2} Offspring Traits  F_{2} Trait Ratios 
Flower color  Violet vs. white  100 percent violet 

3.15:1 
Flower position  Axial vs. terminal  100 percent axial 

3.14:1 
Plant height  Tall vs. dwarf  100 percent tall 

2.84:1 
Seed texture  Round vs. wrinkled  100 percent round 

2.96:1 
Seed color  Yellow vs. green  100 percent yellow 

3.01:1 
Pea pod texture  Inflated vs. constricted  100 percent inflated 

2.95:1 
Pea pod color  Green vs. yellow  100 percent green 

2.82:1 
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. Dominant traits are those that are inherited unchanged in a hybridization. Recessive traits become latent, or disappear, in the offspring of a hybridization. The recessive trait does, however, reappear in the progeny of the hybrid offspring. An example of a dominant trait is the violetflower trait. For this same characteristic (flower color), whitecolored flowers are a recessive trait. The fact that the recessive trait reappeared in the F_{2} generation meant that the traits remained separate (not blended) in the plants of the F_{1} generation. Mendel also proposed that plants possessed two copies of the trait for the flowercolor 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.
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.
Science Practice Connection for AP® Courses
Think About It
Students are performing a cross involving seed color in garden pea plants. Yellow seed color is dominant to green seed color. What F1 offspring would be expected when cross truebreeding plants with green seeds with truebreading plants with yellow seeds? Express the answer(s) as percentage.
Teacher Support
This question is an application of Learning Objectives 3.14 and Science Practice 2.2 because students are applying a mathematical routine (probability) to determine a Mendelian pattern of inheritance.
Possible answer:
Probability Basics
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 expected to occur by the number of times that it could occur. Empirical probabilities come from observations, like those of Mendel. 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 of a genetic event is a round seed produced by a pea plant. In his experiment, Mendel demonstrated that the probability of the event “round seed” occurring was one in the F_{1} offspring of truebreeding parents, one of which has round seeds and one of which has wrinkled seeds. When the F_{1} plants were subsequently selfcrossed, the probability of any given F_{2} offspring having round seeds was now three out of four. In other words, in a large population of F_{2} 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.
The Product Rule and Sum Rule
Mendel demonstrated that the peaplant characteristics he studied were transmitted as discrete units from parent to offspring. As will be discussed, 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. For instance, 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.
The product rule of 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 sixsided die (D) and flipping a penny (P) at the same time. The die may roll any number from 1–6 (D_{#}), whereas the penny may turn up heads (P_{H}) or tails (P_{T}). 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.
Twelve Equally Likely Outcomes of Rolling a Die and Flipping a Penny  

Rolling Die  Flipping Penny 
D_{1}  P_{H} 
D_{1}  P_{T} 
D_{2}  P_{H} 
D_{2}  P_{T} 
D_{3}  P_{H} 
D_{3}  P_{T} 
D_{4}  P_{H} 
D_{4}  P_{T} 
D_{5}  P_{H} 
D_{5}  P_{T} 
D_{6}  P_{H} 
D_{6}  P_{T} 
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_{2}) x (P_{H}) = (1/6) x (1/2) or 1/12 (Table 12.3). Notice the word “and” in the description of the probability. The “and” 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 (for example, yellow and round) in the F_{2} progeny is the product of the probabilities of having the dominant trait for each characteristic, as shown here:
On the other hand, 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. Notice the word “or” in the description of the probability. The “or” indicates that you should apply the sum rule. In this case, let’s 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_{H}) and the quarter may be tails (Q_{T}), or the quarter may be heads (Q_{H}) and the penny may be tails (P_{T}). Either case fulfills the outcome. By the sum rule, we calculate the probability of obtaining one head and one tail as [(P_{H}) × (Q_{T})] + [(Q_{H}) × (P_{T})] = [(1/2) × (1/2)] + [(1/2) × (1/2)] = 1/2 (Table 12.3). You should also notice that we used the product rule to calculate the probability of P_{H} and Q_{T}, and also the probability of P_{T} and Q_{H}, 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_{2} generation of a dihybrid cross:
The Product Rule and Sum Rule  

Product Rule  Sum Rule 
For independent events A and B, the probability (P) of them both occurring (A and B) is (P_{A} × P_{B})  For mutually exclusive events A and B, the probability (P) that at least one occurs (A or B) is (P_{A} + P_{B}) 
To use probability laws in practice, it is necessary to 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 calculate the probabilities of the traits appearing in his F_{2} generation. As you will learn, this discovery meant that when parental traits were known, the offspring’s traits could be predicted accurately even before fertilization.