LS1b08_1_Hartl Probability

# LS1b08_1_Hartl Probability - Probability in Genetic...

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Probability in Genetic Analysis Daniel L. Hartl Harvard University M ENDEL’S LAWS of genetic transmission are fundamentally laws of chance (proba- bility). He surpassed any of his contemporaries in understanding that his principles of inheritance accounted for the different types of progeny he observed, as well as for the ratios in which they were found. No discoveries in genetics made since Mendel’s time have undermined the fundamental role of chance in heredity that he was the first to recog- nize. To understand Mendelian genetics, we therefore need to understand the elementary principles of probability. Every problem in probability begins with an experiment, which may be real or imaginary. In genetics, the experiment is typically a cross. Associated with the experiment is a set of possible outcomes of the experiment. In genetics the pos- sible outcomes are typically genotypes or phenotypes. The possible outcomes are called elementary outcomes. They are elementary outcomes in the sense that none of them can be reduced to combinations of the others. In our applications of probability, the number of elementary outcomes is often relatively small, or in any case can be enumerated. The principles of probability can also deal with conceptual experiments in which there are an infinite number of elementary outcomes, but this requires some technicalities that are not essential for present purposes. Each elementary outcome is assigned a probability that is proportional to its likeli- hood of occurrence. Probabilities always conform to two fundamental rules. First, the probability of each elementary outcome must be a nonnegative number between 0 and 1, and may actually equal 0 or 1. An elementary outcome with a probability of 0 cannot occur, and one with a probability of 1 must occur. The second rule is that the sum of the probabilities of all the elementary outcomes must equal 1. This rule means that some one of the elementary outcome must occur. These two rules also handle an annoying question often asked in regard to a coin toss: What happens if it lands on its edge? The answer is that this elementary outcome is assigned a probability of 0, so we need not bother with it. Here it will be helpful to consider a specific example. Consider a conceptual experi- ment whose outcomes consist of all possible combinations of girls (G) and boys (B) among four offspring produced by a mating between a normal female and a normal male. These are shown in Figure 1A, where the offspring are listed in order of birth from left to right. Each of the elementary outcomes is equally likely, and so the probability of each ele- mentary outcome is assigned a value of 1/16. Note that there are six types of sibships consisting of exactly two girls and exactly two boys. This is because there are six possible orders for the births, namely BBGG, BGBG, GBBG, BGGB, GBGB, and GGBB.

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