What is the probability that two heterozygous individuals will produce a child that is recessive?

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.

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.

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.

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.

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.

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

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.

Probability and statistics are closely related and have important applications in science. In the same way, probability and genetics are also very closely related! Incidences of genetically inherited traits are just instances of probability.

Common Terminology

Alleles: A person's characteristics are the consequence of how the genetic material is paired. This genetic material is known as alleles. It is the framework of these alleles that determines the traits of an individual, like hair color, eye color, height, etc. Certain alleles are dominant, and certain are regressive. A person with one or two dominant alleles exhibits the dominant trait. And a person who has two recessive alleles displays the recessive characteristic.

Genotype: The specific pairing of alleles that an individual inherits that code for a specific trait, is known as a genotype.

Phenotype: The actual physical trait or appearance is known as the phenotype - meaning regardless of the actual genotype, the phenotype is the resulting characteristic of that genotype.

Confusing, right? Let’s look at an example. What determines the hair color of an individual?

• There's a dominant allele ‘B’ that represents black hair color, and there's a recessive allele ‘b’ that represents brown hair color.
• Persons whose alleles are paired as BB or Bb both have black hair. The person has black hair even if the pairing of alleles is different!
• In this case, the particular pair of alleles is known as the person's genotype (the collection of genes an individual has imbibed). In order for an individual to have brown hair, they’d need to have a bb genotype. Black hair would be the phenotype and Bb or BB would be the genotype.
• So if you think about it, there are two genotypes for the phenotype of black hair. However, only people who have bb pairing will have brown hair.

Composition of Genotypes

Now let us talk about the composition of genotypes. A genotype like BB or bb has alleles identical. When a person has this type of genotype, he is known as homozygous. When a genotype is like Bb, the alleles will be different from each other. When an individual has this kind of pairing, this case will be heterogeneous.

Parents and Descendants

Each of the parents possesses a pair of alleles. The parents individually give one of these alleles to their progeny. This is how the descendant derives the pair of its alleles. When we know the parents' genotype, we can expect the genotype and phenotype of the descendant. It has been observed that 50% of each parent's alleles are carried forward to the descendant.

Now, let's again talk about the case of hair color. Both the parents have black hair, have genotype Bb and are heterozygous; there's a chance that each of the parents has a 50% chance of passing down the allele B (which is dominant). There's also a 50 % chance of passing down allele b (recessive).

Below is a gene and allele diagram for better understanding:

Punnett squares

The enumeration above can be more concisely shown by employing a Punnett square. Representation like this is named after Reginald C. Punnett.

What is Punnett square?

A Punnett square comprises a tabular representation that registers all the possible genotypes of the descendant. The parents' genotypes are usually marked outside the Punnett square. The input of each cell in the table is decided by observing the alleles in the column and row of that input.

Let’s put together Punnett square examples for all probable conditions of a single characteristic.

When both parents are homozygous:

All the descendants will possess a matching genotype when both mother and father are homozygous. Let's study the cross between BB and bb. In the tabular presentation below, the parents are represented in bold.

Now all the descendants are heterogeneous, with the genotype of Bb. What's a monohybrid cross Punnett square? This is a tabular representation of a genic combination between two persons who have homozygous genotypes (genotypes having recessive alleles or completely dominant alleles).

When one parent is homozygous:

When one parent is homozygous, the other parent is heterozygous. The Punnett square from one homozygous parent and one heterozygous parent is below.

When the parent is homozygous and has two dominant alleles, all the descendants will have the same dominant trait phenotype. To put it another way, there is a ten out of ten possibility that when a descendant has a pairing of that nature, this pairing will show the dominant phenotype. Now we can also study that there is a possibility that the homozygous parent has two recessive alleles. Now half of the descendants will show the recessive trait with genotype bb. The other half will show the dominant trait, with heterozygous genotype Bb.

When both parents are heterozygous:

When both the mother and the father are heterozygous, both parents will possess the same genotype consisting of one dominant allele and one recessive allele. Based on the above circumstances, the Punnett Square is represented below. Here we see three ways in which an offspring can exhibit a dominant trait, and there is one way for a recessive trait. This means a 25% chance that an offspring will have a recessive trait and a 75% chance that an offspring will have the dominant trait.

What is a dihybrid Punnett square?

In this case, both parents are heterozygous, and one allele for each characteristic shows absolute dominance. This means that both parents have recessive alleles but exhibit the dominant phenotype.

What is a trihybrid cross Punnett square?

A trihybrid cross is a type of Punnett square generated for 3 traits. This kind of Punnett square is a table of 64 boxes, created with the combinations of 6 mother's and 6 father's alleles.

Conclusion:

• Punnett square is useful in calculating the probabilities of offspring with certain genetic characteristics.
• The frame of a Genetic material known as Alleles determines the traits of an individual.
• When a person has Identical Alleles, he is known as homozygous, and when the alleles are different, he is heterogeneous.

FAQs:

1. How do Punnett squares predict the probability?

A Punnett square predicts the percentages of phenotypes in the descendant by studying across from recognized genotypes of both parents.

2. How do you find the probability of offspring using a Punnett square?

Find out the total number of boxes in the particular Punnett Square. This will provide the total figure of offspring predicted. Now you divide the (number of occurrences of the phenotype) by (the total number of offspring). You have to multiply the number from step 4 by 100, and you will obtain your percent.

3. How do you find the probability of an offspring?

You have to divide the number of boxes having an allele dominant by four. Now you multiply the result by 100. This result gives you the probability that an offspring will have the dominant trait. For example (2/4)*100 = 50, therefore there is a 50% probability of an offspring having brown hair.

4. What is probability and how does it relate to genetics?

Probability is a technique applied to predict the possibilities of indefinite results. It is significant for the domain of genetics. It is applied to tell the traits concealed in the genome by the dominant alleles.

5. What is the probability of a heterozygous offspring?

There is a 50% x 50% = 25% probability that both of the alleles of the offspring are dominant. There is a 50% x 50% = 25% probability that both alleles of the offspring's are recessive. There is a 50% x 50% 50% x 50% = 25% 25% = 50% chance that the offspring is heterozygous.

6. How do you do a 4x4 Punnett square?

Making a 4x4 Punnett Square is quite challenging. However, it becomes easy when we use a dihybrid cross calculator.

We hope you enjoyed studying this lesson and learned something cool about Probability in Inheritance - Punnett Squares! Join our Discord community to get any questions you may have answered and to engage with other students just like you! Don't forget to download our App to experience our fun, VR classrooms - we promise, it makes studying much more fun! 😎

Continue your learning journey by learning more about genetics:

1. Mendelian Genetics 2. Non Mendelian Genetics 3. Population Genetics 4. Nucleic Acids

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