# Probability and Punnett Squares in Genetics

Statistics and probability have many applications to science. One such connection between another discipline is in the field of genetics. Many aspects of genetics are really just applied probability. We will see how a table known as a Punnett square can be used to calculate the probabilities of offspring having particular genetic traits.

## Some Terms from Genetics

We begin by defining and discussing some terms from genetics that we will use in what follows. A variety of traits possessed by individuals are the result of a pairing of genetic material. This genetic material is referred to as alleles. As we will see, the composition of these alleles determines what trait is exhibited by an individual.

Some alleles are dominant and some are recessive. An individual with one or two dominant alleles will exhibit the dominant trait. Only individuals with two copies of the recessive allele with exhibit the recessive trait. For example, suppose that for eye color there is a dominant allele B that corresponds to brown eyes and a recessive allele b that corresponds to blue eyes. Individuals with allele pairings of BB or Bb will both have brown eyes. Only individuals with pairing bb will have blue eyes.

The above example illustrates an important distinction. An individual with pairings of BB or Bb will both exhibit the dominant trait of brown eyes, even though the pairings of alleles are different. Here the specific pair of alleles are known as the genotype of the individual. The trait that is displayed is called the phenotype. So for the phenotype of brown eyes, there are two genotypes. For the phenotype of blue eyes, there is a single genotype.

The remaining terms to discuss pertain to the compositions of the genotypes. A genotype such as either BB or bb the alleles are identical. An individual with this type of genotype is called homozygous. For a genotype such as Bb the alleles are different from one another. An individual with this type of pairing is called heterozygous.

## Parents and Offspring

Two parents each have a pair of alleles. Each parent contributes one of these alleles. This is how how the offspring obtains its pair of alleles. By knowing the genotypes of the parents, we can predict the probability what the offspring's genotype and phenotype will be. Essentially the key observation is that each of a parent's alleles has the probability of 50% of being passed down to an offspring.

Let's go back to the eye color example. If a mother and father are both brown eyed with heterozygous genotype Bb, then they each have probability of 50% of passing on the dominant allele B and a probability of 50% of passing on the recessive allele b. The following are the possible scenarios, each with probability of 0.5 x 0.5 = 0.25:

• Father contributes B and mother contributes B. The offspring has genotype BB and phenotype of brown eyes.
• Father contributes B and mother contributes b. The offspring has genotype Bb and phenotype of brown eyes.
• Father contributes b and mother contributes B. The offspring has genotype Bb and phenotype of brown eyes.
• Father contributes b and mother contributes b. The offspring has genotype bb and phenotype of blue eyes.

## Punnett Squares

The above listing can be more compactly demonstrated by using a Punnett square. This type of diagram is named after Reginald C. Punnett. Although it can be used for more complicated situations than the ones that we will consider, other methods are easier to use.

A Punnett square consists of a table listing all of the possible genotypes for offspring. This is dependent upon the genotypes of the parents being studied. The genotypes of these parents are typically denoted on the outside of the Punnett square.  We determine the entry in each cell in the Punnett square by looking at the alleles in the row and column of that entry.

In what follows we will construct Punnett squares for all possible situations of a single trait.

## Two Homozygous Parents

If both parents are homozygous, then all of the offspring will have an identical genotype. We see this with the Punnett square below for a cross between BB and bb.  In all that follows the parents are denoted with bold.

All of the offspring are now heterozygous, with genotype of Bb.

## One Homozygous Parent

If we have one homozygous parent, then the other is heterozygous. The resulting Punnett square is one of the following.

Above if the homozygous parent has two dominant alleles, then all of the offspring will have the same phenotype of the dominant trait. In other words, there is a 100% probability that an offspring of such a pairing will exhibit the dominant phenotype.

We could also consider the possibility that the homozygous parent possesses two recessive alleles.  Here if the homozygous parent has two recessive alleles, then half of the offspring will exhibit the recessive trait with genotype bb. The other half will exhibit the dominant trait but with heterozygous genotype Bb. So in the long run, 50% of all offspring from these types of parents

## Two Heterozygous Parents

The final situation to consider is the most interesting.  This is because the probabilities that result.  If both parents are heterozygous for the trait in question, then they both have the same genotype consisting of one dominant and one recessive allele.

The Punnett square from this configuration is below.  Here we see that there are three ways for an offspring to exhibit a dominant trait and one way for recessive.  This means that there is a 75% probability that an offspring will have the dominant trait and a 25% probability that an offspring will have a recessive trait.

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