Probability and the product rule

Plant breeders use genetics to plan crops, and genetic counselors use it to tell families the chance of passing on a disease. Neither can promise an outcome. What they can give is a probability, and a little arithmetic lets you predict crosses too complicated for one Punnett square.

What a probability is

The probability of an outcome is the chance it will happen. It equals the number of ways the outcome can occur divided by the total number of equally likely outcomes:

P = (ways the outcome can occur) ÷ (total possible outcomes)

A coin has two equally likely sides, so the probability of heads is 1/2, or 50%. Drawing one particular card from a full deck is 1/52.

A Punnett square is a probability tool in disguise. In a Tt × Tt cross, the four boxes are four equally likely outcomes, and one of them is tt. So the probability that an offspring is short is 1/4, or 25%.

Small samples stray from the odds

A probability predicts what happens on average, not in any one case. Each coin toss, and each child, is an independent event: its outcome doesn’t depend on what came before. Toss a coin six times and you might easily get four heads and two tails instead of three and three.

The gap between what you observe and what you expect, caused purely by chance, is random sampling error. The smaller the sample, the larger it tends to be. Toss the coin 1000 times and the share of heads will land close to 50%. This is one reason human pedigrees, with their small families, give less certain answers than Mendel’s counts of hundreds of pea plants.

Check yourselfA Tt × Tt cross produces only 8 offspring, and 4 of them are short. Does that disprove the expected 3:1 ratio?

No. With so few offspring, random sampling error can easily push the result away from 3:1. A much larger sample would be needed to test the ratio.

The product rule

To find the chance that several independent events all happen, use the product rule: the probability that two or more independent events all occur is the product of their individual probabilities.

The chance of two heads in two tosses is 1/2 × 1/2 = 1/4.

Splitting a cross into one gene at a time

The product rule also handles crosses involving several genes, as long as the genes assort independently. Instead of drawing one huge Punnett square, work out each gene separately, then multiply.

The same method gives whole phenotype ratios. In RrYy × rrYy, seed shape (Rr × rr) is 1/2 round and 1/2 wrinkled, and seed color (Yy × Yy) is 3/4 yellow and 1/4 green. Multiplying gives 3/8 round yellow, 1/8 round green, 3/8 wrinkled yellow, and 1/8 wrinkled green, a 3:1:3:1 ratio.

Check yourselfTwo pea plants, AaBb × AaBb, are crossed. The genes assort independently. What is the probability that an offspring is aabb?

1/16. Each gene alone gives the homozygous recessive with probability 1/4, so 1/4 × 1/4 = 1/16.