Probability Theory Help
Law of Products
There are two laws of probability that are used for genetic analysis. The first law, the law of products (or product rule), is used to predict the probability of two or more independent events occurring together. Two or more events are said to be independent if the occurrence or nonoccurrence of any one of them does not affect the probability of occurrence of any of the others.When two independent events occur with the probabilities p and q, respectively, then the probability of their join to ccurrence is (p)(q).That is, the combined probability is the product of the probabilities of the independent events. If the word "and" is used or implied in the phrasing of a problem solution, a multiplication of independent probabilities is usually required.
EXAMPLE 2.21 Theoretically, there is an equal opportunity for a tossed coin to land on either heads or tails. Let p = probability of heads = 1/2, and q = probability of tails = 1/2. In two tosses of a coin the probability of two heads appearing (i.e., a head on the first toss and a head on the second toss) is p × p = p2 = (1/2)2 = 1/4.
EXAMPLE 2.22 In test crossing a heterozygous black guinea pig (Bb × bb), let the probability of a black (Bb) offspring be p = 1/2 and of a white (bb) offspring be q = 1/2. The combined probability of the first two offspring being white (i.e., the first offspring is white and the second offspring is white) = q × q = q2 = (1/2)2 = 1/4.
Law of Sums
There is only one way in which two heads may appear in two tosses of a coin, i.e., heads on the first toss and heads on the second toss. The same is true for two tails. There are two ways, however, to obtain one head and one tail in two tosses of a coin. The head may appear on the first toss and the tail on the second or the tail may appear on the first toss and the head on the second. If a head occurs on the first toss and on the second toss, a tail cannot also occur on the second toss. Therefore, these events are mutually exclusive. Mutually exclusive events are those in which the occurrence of any one of them excludes the occurrence of the others. The law of the sum (or sum rule) is used to predict the probability that two mutually exclusive events will occur and states that the probability is the sum of their individual probabilities. The word "or" is usually required or implied in the phrasing of problem solutions involving mutually exclusive events, signaling that an addition of probabilities is to be performed. That is, whenever alternative possibilities exist for the satisfaction of the conditions of a problem, the individual probabilities are combined by addition.
EXAMPLE 2.23 In two tosses of a coin, there are two ways to obtain a head and tail.
EXAMPLE 2.24 In test crossing heterozygous black guinea pigs (Bb × bb), there are two ways to obtain one black (Bb) and one white (bb) offspring in a litter of two animals. Let p = probability of black = 1/2 and q = probability of white = 1/2.
Many readers will recognize that the application of the above two laws for combining probabilities is the basis of the binomial distribution, which will be considered in the next section
Practice problems for these concepts can be found at:
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