Genetic probability calculator.

Product rule, sum rule and several children

The product rule says the probability of two independent events occurring together is found by multiplying their individual probabilities. The sum rule says that the probability of one or the other of two mutually exclusive events is the sum of their probabilities (OpenStax Biology 2e, 12.1). Each child in a family is an independent event with the same per-child chance, so the chance that exactly k of n children show a trait is the binomial probability: (number of birth orders) × p^k × (1 − p)^n−k (Wikipedia, Binomial distribution). "At least one" is 1 minus the chance that none do.

Worked example. Aa × Aa gives a 3/4 chance of the dominant phenotype per child. For 4 children, exactly 3 dominant has 4 birth orders, each (3/4)³ × (1/4) = 0.1055, so 0.4219 (42.2%), which is 27/64. For two independent traits at 3/4 each, the product rule gives 0.5625 (9/16), as in OpenStax's dihybrid example. Two mutually exclusive outcomes of 3/16 each, A_bb or aaB_, add to 0.375 (3/8). A test cross heterozygote gives a 1/2 chance per child, so all 4 children showing the dominant trait has a 0.0625 chance (1/16).