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OpenStax Biology 2e

Mendel demonstrated that pea plants transmit characteristics 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 yellow:green seeds (ignoring seed texture) and a 3:1 ratio of wrinkled:round 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 six-sided 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, and each event is expected to occur with equal probability.

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. 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. 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:

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 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.

Source:

Clark, M., Douglas, M., Choi, J. *Biology 2e*. Houston, Texas: OpenStax. Access for free at: https://openstax.org/details/books/biology-2e