AP Biology · Unit 5 · Topic 5.3
A Punnett square is not a rule to memorise, it is bookkeeping for segregation. Build the square, count the boxes, and watch 9:3:3:1 turn out to be two 3:1s multiplied.
Learning objective 5.3.A Explain the inheritance of traits using Mendel's laws.
A Punnett square is not a rule you memorise. It is a list of one parent's gametes against the other's, and the ratio is the result of counting. Change the parents and watch the ratio follow.
Set both parents to Aa and look at the four boxes rather than the ratio. Three of them contain at least one A, so three of them show the dominant phenotype. The 3:1 is not a law of genetics; it is what happens when you count four boxes made from two alleles each.
Because there are two ways to be a heterozygote and only one way to be either homozygote. The A can come from parent 1 with the a from parent 2, or the other way round, and both produce Aa. Those are two separate boxes.
This is the same reason a pair of dice gives seven more often than two: more of the possible combinations land on it.
Try it: set parent 1 to Aa and parent 2 to aa, then change parent 1 to AA. The phenotype ratio changes, so the cross has told you something. Now set parent 2 to AA instead and try the same two options for parent 1. Both give all-dominant offspring, and you have learned nothing.
Everything the square does follows from that. The row and column headers are a parent's gametes, and there are two of them because the two alleles separate.
Two genes, each with complete dominance, on different chromosomes. Each parent is AaBb and so makes four gamete types. Pick a phenotype class to see which boxes belong to it.
Each dihybrid class is the product of the two monohybrid classes it is made of. Nothing new is being asserted about two genes that was not already true of one — which is exactly the claim independent assortment makes.
Ignore the second gene entirely and count the sixteen boxes by the first gene only. Then do it the other way round. Both give 12:4, which is 3:1. Each gene is behaving exactly as it did in its own square, undisturbed by the other — and that is what makes the multiplication legitimate.
Because the classes multiply, you never have to draw sixteen boxes. The chance of an offspring being recessive for both genes is ¼ × ¼ = 1/16; dominant for the first and recessive for the second is ¾ × ¼ = 3/16. For three genes the square would have 64 boxes and the multiplication still takes one line.
The shortcut depends on the genes assorting independently. Two genes close together on the same chromosome do not, and the observed ratio then departs from 9:3:3:1 — which is exactly the departure the next tab tests for.
Real offspring never match a prediction exactly. Chi-square measures how far off they are and asks whether a departure that large could plausibly come from chance alone. Pick a dataset and work through it.
| Class | Observed | Expected | O − E | (O − E)² / E |
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A 3:1 cross has two phenotype classes and therefore one degree of freedom, whether you counted 20 offspring or 2000. A 9:3:3:1 cross has four classes and three degrees of freedom. Step between the datasets above and watch the degrees of freedom change with the number of classes and not with the sample size.
Degrees of freedom does not depend on the sample size, but the statistic does. The same proportions counted ten times over give a chi-square ten times larger, because a departure of a given proportion is far less likely to arise by chance in a large sample. This is why a result you could shrug off in a small experiment demands an explanation in a big one.
“The data are consistent with a 3:1 ratio” is a defensible sentence. “The chi-square test proves the trait is inherited in a 3:1 ratio” is not, and it costs marks. The test can only ever tell you whether you have grounds to abandon the hypothesis you started with.
Answer these in your own words before you open the answers.
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| Cross | Genotypes | Genotype ratio | Phenotype ratio |
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Every row here was produced by building the square and counting, not by looking a ratio up. If you can rebuild any row from the parents alone, you can handle a cross you have never seen.