What this calculator does
A single Aa × Aa cross gives a one in four chance of the recessive phenotype. Across several genes assorting independently, the question becomes how many of them come out recessive, which is a binomial problem: each gene is an independent trial with a probability of 0.25.
For two genes the probabilities work out at 9/16 for neither recessive, 6/16 for exactly one and 1/16 for both, which is the classic 9:6:1 pattern behind the dihybrid ratio. Drawing a 16-square Punnett grid gives the same answer more slowly, and becomes impractical beyond two or three genes.
The formula
Each gene pair is treated as an independent trial with a one in four chance of producing the recessive phenotype. The probability of exactly r recessive traits out of n genes is the binomial expression: the number of ways to choose r from n, multiplied by 0.25 to the power r, multiplied by 0.75 to the power n minus r. The expected offspring count multiplies that probability by the total.
| Term | Meaning |
|---|---|
| Independent assortment | Mendel second law: genes on different chromosomes segregate independently, which is what makes the binomial treatment valid. |
| Monohybrid ratio | The 3:1 phenotype ratio from a single Aa × Aa cross, equivalent to a 0.25 recessive probability. |
| Binomial coefficient | The number of ways r recessive traits can be chosen from n genes, written C(n, r). |
The inputs explained
| Field | What to enter |
|---|---|
| Heterozygous gene pairs (Aa × Aa) | How many gene pairs are in the cross, each assumed to be Aa × Aa. |
| Number showing recessive phenotype | How many of them you want to come out recessive. |
| Offspring to project onto | Total offspring, used only to convert the probability into an expected count. |
When to use it
Solving a dihybrid or trihybrid problem
Punnett grids grow as 4 to the power n, so three genes needs a 64-square grid and four needs 256. The binomial gives the same answers directly.
Predicting a breeding outcome
Where several independent recessive traits are being tracked, the probability of a particular combination is what determines how many offspring are needed to have a fair chance of getting it.
Checking a Punnett square
Working the probability independently is a quick check on a grid that has been filled in by hand, where miscounting squares is easy.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What are the odds across two gene pairs?
The probability of each possible number of recessive traits across two independent genes.
| Recessive traits wanted | Probability | Expected offspring out of 16 | P(all recessive) |
|---|---|---|---|
| 0 | 56.3% | 9.00 | 6.25% |
| 1 | 37.5% | 6.00 | 6.25% |
| 2 | 6.25% | 1.00 | 6.25% |
Questions
Why is the probability of a recessive trait 1 in 4?
Because an Aa × Aa cross produces AA, Aa, aA and aa in equal proportions, and only aa shows the recessive phenotype. That is one outcome in four, which is the 3:1 monohybrid ratio stated the other way round.
When does independent assortment not apply?
When the genes are linked, meaning they sit close together on the same chromosome and tend to be inherited together. The binomial treatment then gives wrong answers, and the degree of error depends on how close the loci are and how often recombination separates them.
Why use a binomial instead of a Punnett square?
Because Punnett squares scale badly. One gene needs 4 squares, two need 16, three need 64 and four need 256. The binomial gives the same answer in one step at any number of genes, and is far less error-prone.
Does this work for crosses other than Aa x Aa?
No, it assumes both parents are heterozygous at every locus, which fixes the per-gene recessive probability at 0.25. Other cross types give different per-gene probabilities and would need those substituted into the binomial.
For expected genotype frequencies in a whole population, see the Hardy-Weinberg calculator. For allele frequencies from observed counts, see the allele frequency calculator.