A Punnett square is a picture of probability. For bigger questions, such as three genes at once or several children in a row, drawing squares gets slow. Two rules from Compound Probability do the work faster.
Key idea
Product rule: for independent events, the chance that both happen is the product. Sum rule: for outcomes that cannot happen together, the chance of one or the other is the sum.
Why it works
Genes on different chromosomes sort independently in meiosis (see How Meiosis Makes Variation), so what one gene does tells you nothing about another. That is what "independent" means, and it is why you can multiply. A child who is AA cannot also be Aa, so those outcomes are separate and you can add them.
Worked example: The product rule
Parents are AaBb × AaBb, with genes on different chromosomes. What is the chance of an aabb child?
- Gene A alone: Aa × Aa gives aa with chance ¼.
- Gene B alone: Bb × Bb gives bb with chance ¼.
- Both: ¼ × ¼.
Answer: $\frac{1}{16}$
Worked example: The sum rule
From Aa × Aa, what is the chance the offspring is heterozygous?
- Aa can happen two ways: A from the first parent and a from the second, or a first and A second.
- Each way has chance ½ × ½ = ¼.
- Add them: ¼ + ¼.
Answer: ½
Worked example: Both rules together
AaBbCc × AaBbCc. What is the chance of an offspring that shows the dominant A trait, is bb, and shows the dominant C trait?
- Dominant A: ¾ (AA or Aa, sum rule: ¼ + ½).
- bb: ¼.
- Dominant C: ¾.
- Multiply: ¾ × ¼ × ¾ = 9/64.
Answer: $\frac{9}{64}$
Common mistake
Two common errors. Adding when you should multiply: the chance of aa and bb is not ¼ + ¼ = ½; that would make a rare combination common. And the "due" error: if two Aa parents already have three children with the dominant trait, the chance the next one is aa is still ¼. Each child is a new, independent event; the alleles have no memory.
Quick check: a product of chances is never bigger than either chance, and a sum of separate outcomes is never more than 1.