One common application of probability in genetics is the calculation of the probability that a progeny with have a certain genotype (or phenotype). When you are first introduced to this you used a Punnett square which is really just an easy way to visually apply the AND rule and the OR rule to perform the calculation. The AND rule is applied when you multiply the gamete frequencies (e.g. Pr(A AND a) for a single square. The OR rule is applied when you add together the probabilities of each square with the same genotype.

So, from the cross Bb x Bb, the probability of getting a progeny with the dominant phenotype is Prob(B AND b) OR Prob(b AND B), where the male and female gametes are listed respectively, OR Prob (B AND B). Basically, there are two permutations of (B AND b). The answer then is (½ * ½) + (½ * ½) + (½ * ½) = ¾ which is what you get from a Punnett Square.

Once you know the genotype probabilities for a cross (from a Punnett square, or just by becoming familiar with the limited number of possible crosses) you can apply these probabilities when dealing with probability questions about multiple progeny. Common examples for multiple progeny are:

If you perform the cross Ff x Ff and get 2 progeny, what is the probability that they are both ff?

This is an application of the AND rule: Pr(ff AND ff) = ¼ * ¼ = 1/16.

If you perform the cross Ff x Ff and get 2 progeny, what is the probability that either, or both, is ff?

This is an application of the non-exclusive OR rule: ¼ + ¼ - ¼ * ¼ = 7/16

If you perform the cross Ff x Ff and get 3 progeny, what is the probability that only one is ff?

This is an application of the AND rule with permutations (since the order of the progeny is NOT specified): Pr(F_ AND F_ AND ff) = ¾ * ¾ * ¼ = 9/64

With 3 permutations: 3 * 9/64 = 27/64

If you perform the cross Ff x Ff and get 3 progeny, what is the probability that the first two progeny have the dominant phenotype but the third has the recessive phenotype?

This is the same as the preceding question except that you do NOT account for permutations since the order is specified: Pr(F_ AND F_ AND ff) = ¾ * ¾ * ¼ = 9/64

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