C13 Probabilities | 29 | v4 | BIOS Instant Notes in Genetics

ABSTRACT

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Probabilities

When we toss a coin we cannot say which way up it will land because each landing is a random or stochastic event. We can say that it has a probability of 0.5 of being ‘heads’ and 0.5 of being ‘tails’; both outcomes are equally likely. Probability theory tells us how often we can expect something to turn out in a particular way when we cannot predict the outcome of an individual event. In science, our hypothesis (about the mechanism of inheritance) predicts the probabilities of particular results, and allows us to test that hypothesis.

The sums rule (OR)

The sums rule (addition) is applied to combine probabilities of events which are mutually exclusive. A die has six numbered sides and each side has a probability of 1/6 of being on top. The probability of getting a 5 or a 6 (getting one excludes getting the other) is 1/6 1/6 2/6 1/3 P_{(5 or 6)} = p ₍₅₎ + p ₍₆₎.

The products rule (AND)

The products rule (multiplication) is applied to independent events. The probability of a child being a boy is 0.5 and of being a girl is 0.5 (p _(B) = 0.5 and p _(G) = 0.5). These can be multiplied to give the probability of two consecutive babies being boys, which is 0.5 × 0.5² = 0.52 = 0.25 (i.e.1/4).

Calculating probabilities

Independent events are not affected by previous events. If the first three children in a family are boys the fourth still has a probability of 1/2 of being a girl. The product rule is often combined with the sums rule, for example to calculate the probability of two consecutive babies being both boys or both girls: 0.5² _{(both boys)} + 0.5² _{(both girls)} = 0.5 (i.e. 1/2).

Punnett squares can be used to calculate probabilities, but probability paths are less error prone. In a two-factor cross (AABB × aabb), the probability of F2 individuals being homozygous recessive (aa) at the first locus is 1/4 and of being heterozygous (Bb) at the second locus is 1/2, so the probability of being (aaBb) is 1/4 × 1/2 = 1/8. The sum of the end-points of all paths must be 1.

Permutations

Permutations are different ways (arrangements, combinations) which give the same result. There are two permutations by which two babies can be one boy and one girl; the boy can be the first one or the last one. The probability of a boy (0.5) followed by a girl (0.5) is 0.5² = 0.25. This must be multiplied by two to include the probability of a girl followed by a boy, so the probability of a boy and girl in any order is 0.5² × 2 = 0.5. The probabilities of all possible outcomes add up to 1 (unity), 0.25 of two boys, 0.5 of a girl and a boy, and 0.25 of two girls.

The number of permutations is the coefficient of the terms of a binomial expansion: P = (n!/s!t!).

The ‘!’ means factorial, the product of all integers down to 1, so 3! = 3 × 2 × 1 = 6. If the probability of the first outcome is p and of the second is q (where p + q = 1) then the probability of s first events and t second events in n trials (where n = s + t) is P _{(s and t)} = (n!s!t!)p ^s q ^t.

The coefficient (n!/s!t!) is the number or permutation, and p ^s qt is the probability of a specific sequence of events. With n events, each with two possibilities, there are (n+1) numerical categories: if there are three children, zero, one, two, or three could be boys. The coefficients (number of ways of getting) for each category can also be read from the (n+1) row of Pascal's triangle. When there are more than two alternative outcomes for each event (e.g. six numbers on a die) a multinomial expansion must be used.