A Punnett square is a compact way to enumerate possible gamete combinations in a genetic cross. It does not create inheritance probabilities; it displays probabilities that come from gamete production and fertilization.

Consider the cross

$$Aa\times Aa.$$

Each parent produces $A$ and $a$ gametes with probability $1/2$. Place one parent's gamete types across the columns and the other's along the rows:

Each cell represents one joint gamete event. Because the two parental gamete choices are independent here,

$$P(A\text{ from parent 1 and }a\text{ from parent 2})=\frac12\cdot\frac12=\frac14.$$

Counting the cells gives

$$P(AA)=\frac14,$$

$$P(Aa)=\frac24=\frac12,$$

$$P(aa)=\frac14.$$

Phenotype probabilities require a genotype-to-phenotype rule

If $A$ is completely dominant to $a$, then $AA$ and $Aa$ share the dominant phenotype. Therefore

$$P(\text{dominant phenotype})=P(AA)+P(Aa)=\frac14+\frac12=\frac34,$$

and

$$P(\text{recessive phenotype})=\frac14.$$

The addition step combines mutually exclusive genotype outcomes that produce the same phenotype.

Punnett squares are tables of possibilities, not literal families

A $2\times2$ square has four cells, but this does not mean every family of four offspring will contain exactly one $AA$, two $Aa$ and one $aa$ child. Each offspring is a new probabilistic event. The ratios describe expected frequencies over many independent offspring under the model.

Probability reasoning scales better than large squares

Punnett squares are useful for one or two loci, but the underlying probability rules are more general. For example, if an offspring from $Aa\times Aa$ has probability $3/4$ of the dominant phenotype, then the probability that two independent offspring both show it is

$$\left(\frac34\right)^2=\frac9{16}.$$

The core procedure is therefore:

1. determine gamete probabilities from parental genotypes;
2. combine gametes using probability multiplication;
3. group mutually exclusive offspring genotypes when needed using probability addition;
4. translate genotype probabilities into phenotype probabilities only after specifying the inheritance relationship.