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As the binomial is expanded for each value of n, Pascal's triangle is useful in determining the numerical coefficient of each term in the binomial equation. In this triangle, each number is the sum of the two numbers immediately above it.

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A Punnett square can be used to predict the genotype and phenotype ratios of a monohybrid cross

The key step is to line up the gametes along the rows and columns of the square.

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To interpret the Χ² value, first determine the value of the degrees of freedom (df), which is equal to n – 1, where n is the number of different categories into which each datum point may fall. The greater the number of categories, the more deviation is expected as a result of chance.

For the 3:1 ratio, n = 2, so df = 1. For the 9:3:3:1 ratio, df = 3. Χ² can then be determined by looking up the probability value (p) in a table.

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Χ² values greater than those shown at p = 0.05 fail to reject the null hypothesis, while Χ² values less than those at p = 0.05 justify rejecting the null hypothesis. In our example, Χ² = 0.53 for 1 df is converted to a p value between 0.20 and 0.50. The graph in (a) provides an estimated p value of 0.48 by interpolation. In this case, we fail to reject the null hypothesis.

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F₁ and F₂ results of Mendel's dihybrid crosses between yellow, round and green, wrinkled pea seeds and between yellow, wrinkled and green, round pea seeds.

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Independent assortment of two traits results in an equal frequency of the formation of all the possible combinations of gametes, with a predicted 9:3:3:1 dihybrid ratio in the F₂ generation.

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A 9:3:3:1 dihybrid ratio in the F₂ generation is characteristic of the independent assortment of two traits. One method of obtaining all the possible combinations of gamete genotypes is using the FOIL procedure similar to the multiplication of two binomial expressions.

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To generate phenotypic ratios in the F₂, consider the A/a, B/b, and C/c gene pairs separately. For example, for the A/a pair, the F₁ cross is Aa x Aa.

In the F₂, a phenotypic ratio of 3/4 A: 1/4 a, and a genotypic ratio (not shown here) of 1/4 AA:1/2 Aa:1/4 aa is produced, assuming independent assortment.

The same ratios apply to the BB x bb and CC x cc crosses.

The proportions of organisms expressing each phenotypic combination can be predicted by multiplying the individual probabilities, using the product law.

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An individual may possess a maximum of two alleles for each gene, one alleles inherited from each parent.

Some traits, such as flower color in the pea, may have more than two alleles in the population.

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Pedigrees have been very useful in determining how human genetic diseases such as hemophilia are inherited.

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Assuming the two pairs of contrasting traits are inherited independently, the product law predicts that the combined probability of the two phenotypes is equal to the product of their individual probabilities.

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In a AaBBCcDd x AaBBCcDd cross, the heterozygous gene pairs (n) is 3 (because the B genes are not heterozygous), assuming independent assortment. Each parent can produce 8 (2³) different gametes and yield 27 (3³) different genotypes and 6 different phenotypes in the F₂ generation.

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Genes on different chromosomes align randomly during metaphase I, resulting in independent assortment.

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Mendel's pairs of unit factors are really genes located on homologous pairs of chromosomes.

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Members of each pair of homologs separate in meiosis I, resulting in independent segregation.

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A monohybrid cross between tall and dwarf pea plants. A heritable characteristic, such as the height of the plant, is called a phenotype.

The symbols D and d designate the tall and dwarf unit factors (genes), respectively, in the genotypes of mature plants (sporophytes) and gametes.

Sporophytes are shown in rectangles, and gametes are shown in circles.

In the F₁ generation, all of the plants show just one of the two contrasting traits.

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In the F₂ generation, 3/4 of the plants exhibit the same trait as the F₁ generation, and 1/4 exhibit the contrasting trait that disappeared in the F₁ generation.

To explain these results, Mendel proposed the existence of particulate unit factors (genes) for each characteristic (phenotype).

He suggested that these factors serve as the units of heredity and are passed from generation to generation, determining various traits.

The two contrasting traits of the same gene are called alleles. An individual possessing two copies of the same allele is homozygous. An individual possessing different copies of an allele is heterozygous.

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A summary of the seven monohybrid crosses (matings) of the garden pea. Pollen from P1 plants exhibiting one trait was used to fertilize the ova of plants exhibiting the other trait. In the F1 generation, one of the two traits (dominant) was exhibited by all plants. The contrasting trait (recessive) then appeared in about 1/4 of the F2 plants.

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Standard pedigree conventions

Circles represent females and squares designate males. A diamond indicates unknown sex.

Individuals exhibiting the phenotype ("affected") have shaded symbols.

Parents are connected by a single horizontal line and vertical lines lead to their offspring.

If the parents are related (consanguineous, such as first cousins), they are connected by a double line.

Offspring are called sibs (siblings) and are connected by a horizontal sibship line.

Sibs are listed left to right by birth order, labeled with Arabic numerals.

Twins are indicated by diagonal lines stemming from a vertical line connected to the sibship line.

Identical (monozygotic) twins are linked by a horizontal line between the diagonals.

A number in a symbol represents numerous sibs of the same phenotypes.

A proband (p) is the first known affected individual in the pedigree, indicated by an arrow.
An optional diagonal line over the symbol indicates a deceased individual.

An optional dot within an unshaded symbol indicates a heterozygous carrier, if known.

Each generation is indicated by a Roman numeral.

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Autosomal Dominant Trait. All affected individuals have a parent that also expresses the trait. A dominant trait should not skip a generation.

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Autosomal Recessive Trait. Individual II-3 would have to be affected in order to pass the trait to his offspring (III-3 and III-4), but he is not. Therefore II-3 and II-4 must both be heterozygous, and approximately 1/4 of their offspring should be affected, and II-3 and II-4 individuals can be represented by a shaded dot within their symbols.

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Testcross of a single character.
(a)	The tall parent is homozygous.		(b)	The tall parent is heterozygous.
The genotype of each tall parent can be determined by examining the offspring when each is crossed to the homozygous recessive dwarf plant.

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Testcross results of three yellow, round individuals What is the expected F1 phenotypic ratio if the test parent had a genutype of GGWW?
If the test parent had a genotype of GGWw, the F1 would show a genotype ratio of 1/2 GgWw and 1/2 Ggww.		If the test parent had a genotype of GgWw, the F1 would show a genotype ratio of 1/4 GgWw, 1/4 Ggww, 1/4 ggWw, 1/4 ggww.		If the test parent had a genotype of GgWW, the F1 would show a genotype ratio of 1/2 GgWw and 1/2 ggWw.

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In a trihybrid cross between AABBCC and aabbcc, all F₁ offspring are AaBbCc, and can produce 8 different gametes.

Instead of drawing a Punnett square with 64 combinations, a forked-line method can be used to calculate the predicted F₂ ratios.