Example configurations

Number of genes

From 1 to 4 biallelic genes can be represented in diploid organisms.

Letter for each gene

Gene 1

Gene 2

Parental genotypes

Parent 1

Gene 1

Gene 2

Parent 2

Gene 1

Gene 2

Phenotype distribution

Punnett square

Genotype combinations between Parent 1 and Parent 2 gametes
P₁ × P₂	AB	Ab	aB	ab
AB	AABB — A- B-	AABb — A- B-	AaBB — A- B-	AaBb — A- B-
Ab	AABb — A- B-	AAbb — A- bb	AaBb — A- B-	Aabb — A- bb
aB	AaBB — A- B-	AaBb — A- B-	aaBB — aa B-	aaBb — aa B-
ab	AaBb — A- B-	Aabb — A- bb	aaBb — aa B-	aabb — aa bb

Probability details

Phenotype distribution

Phenotype	Probability	Exact fraction
A- B-	56.25%	9/16
A- bb	18.75%	3/16
aa B-	18.75%	3/16
aa bb	6.25%	1/16

Genotype distribution

Genotype	Probability	Exact fraction
AABB	6.25%	1/16
AABb	12.5%	1/8
AAbb	6.25%	1/16
AaBB	12.5%	1/8
AaBb	25%	1/4
Aabb	12.5%	1/8
aaBB	6.25%	1/16
aaBb	12.5%	1/8
aabb	6.25%	1/16

Results

Genotypic classes

Phenotypic classes

Most probable phenotype

A- B-

Maximum probability: 56.25%

Offspring homozygous for every gene

25%

1/4

Phenotypic ratio

A- B-9

A- bb3

aa B-3

aa bb1

9:3:3:1

Fundamentals & Explanation

From alleles to phenotypes

A locus is the position considered for a gene. Two variants, or alleles, are represented here: an uppercase letter such as $A$ and its lowercase counterpart, $a$ . A diploid individual carries two alleles at each locus and transmits one of them in each gamete.

Under complete dominance, $AA$ and $Aa$ produce the same dominant phenotype; $aa$ produces the recessive phenotype. Dominance describes the phenotype of the heterozygote. It does not indicate that an allele is more frequent, more advantageous, or molecularly inactive.

Relationship between genotype and phenotype under complete dominance
Genotype	Composition	Phenotype
AA	Homozygous	Dominant
Aa	Heterozygous	Dominant
aa	Homozygous	Recessive

The notation $A-$ groups $AA$ and $Aa$ when only the phenotype is being considered.

A monohybrid cross step by step

In the cross $Aa\times Aa$ , each parent produces $A$ and $a$ gametes with probability $1/2$ . The four gamete combinations have equal probability:

Punnett square for Aa by Aa
Aa × Aa	A	a
A	AA	Aa
a	Aa	aa

P(AA)=\frac14,\quad P(Aa)=\frac12,\quad P(aa)=\frac14

Grouping $AA$ and $Aa$ by phenotype gives:

P(A-)=\frac34,\qquad P(aa)=\frac14

The theoretical phenotypic ratio is $3:1$ . A finite set of offspring may show different frequencies because of sampling variation.

From one locus to several

When loci assort independently, the probability of a specific combination is the product of the required probability at each locus:

P(E_1\cap\cdots\cap E_n)=\prod_{i=1}^{n}P(E_i)

For $AaBb\times AaBb$ , each locus contributes a probability of $3/4$ for the dominant phenotype and $1/4$ for the recessive phenotype. For example, $P(A-\,B-)=(3/4)(3/4)=9/16$ .

Phenotypic distribution of the dihybrid cross AaBb by AaBb
Phenotype	Probability	Relative weight
A- B-	9/16	9
A- bb	3/16	3
aa B-	3/16	3
aa bb	1/16	1

The general expression for a cross that is heterozygous at every locus is:

\prod_{i=1}^{n}\left(\frac34D_i+\frac14r_i\right)

The subscripts identify different loci. In this particular cross, up to $3^n$ genotypic classes and $2^n$ phenotypic classes can occur. Other parental genotypes produce fewer classes with nonzero probability.

Reading the results

Genotype distribution: Probability of each allele combination, such as $AABB$ , $AaBb$ , or $aabb$ .
Phenotype distribution: Grouping of genotypes that produce the same phenotype under complete dominance.
Ratio: Simplified relative weights. $9:3:3:1$ corresponds to probabilities $9/16$ , $3/16$ , $3/16$ , and $1/16$ ; it is not an observed count.
Homozygous for every gene: Probability that the offspring carries two identical alleles in every active gene, without requiring them all to be dominant or all recessive.

Segregation and independent assortment

The law of segregation states that the two alleles at a locus separate during gamete formation. In a heterozygote $Aa$ , each allele is transmitted with probability $1/2$ under equal segregation.

Independent assortment allows probabilities to be multiplied across unlinked loci. Physically close loci can be inherited in a correlated manner; that case requires haplotype phase and recombination fraction.

Assumptions and limits

Each gene is biallelic and represented by two alleles in a diploid organism.
The uppercase allele is defined as completely dominant over the lowercase allele.
In heterozygotes, each allele has probability 1/2 of entering a gamete.
Loci assort independently, and all zygotes have equal viability.
The probabilities are theoretical; no random sample of offspring is generated.

Linkage, incomplete dominance, codominance, multiple alleles, epistasis, penetrance, and differential viability are outside this scope.

References

OpenStax Biology 2e — 12.3 Laws of Inheritance. Segregation, independent assortment, and mono- and dihybrid crosses.
Nature Scitable — Gregor Mendel and the Principles of Inheritance. Archived historical resource.

Mendelian cross simulator

Letter for each gene

Parental genotypes

Phenotype distribution

Punnett square

Phenotype distribution

Genotype distribution

Results

Phenotypic ratio

Fundamentals & Explanation

From alleles to phenotypes

A monohybrid cross step by step

From one locus to several

Reading the results

Segregation and independent assortment

Assumptions and limits

References

From alleles to phenotypes

A monohybrid cross step by step

From one locus to several

Reading the results

Segregation and independent assortment

Assumptions and limits

References