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Twelve Equally Likely Outcomes of Rolling a Die and Flipping a Penny | |
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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 ( [link] ). 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 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 ( [link] ). 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 just one dominant trait in the F _{2} generation of a dihybrid cross:
The Product Rule and Sum Rule | |
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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.
Working with garden pea plants, Mendel found that crosses between parents that differed by one trait produced F _{1} offspring that all expressed the traits of one parent. Observable traits are referred to as dominant, and non-expressed traits are described as recessive. When the offspring in Mendel’s experiment were self-crossed, the F _{2} offspring exhibited the dominant trait or the recessive trait in a 3:1 ratio, confirming that the recessive trait had been transmitted faithfully from the original P _{0} parent. Reciprocal crosses generated identical F _{1} and F _{2} offspring ratios. By examining sample sizes, Mendel showed that his crosses behaved reproducibly according to the laws of probability, and that the traits were inherited as independent events.
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. The use of the word “and” suggests the appropriate application of the product rule. To find the probability of two or more events occurring in combination, apply the sum rule and add their individual probabilities together. The use of the word “or” suggests the appropriate application of the sum rule.
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