# Probabilities in Genetics

• Cell Biology • Ecology • Infection • Organization Genetics involves probabilities and some math, but don't worry, mathifies analysis We can use from simple garden pea genetic disorders passed down in families. There are two laws that help us calculate probabilities in genetics: the sum law and the product law.

The sum law, also known as the "OR" rule, helps us find the probability of two or more events occurring as long as they are mutually exclusive. This means that either one event can happen, or the other can happen, but not both. To use the sum law, we simply add the probabilities of each individual event.

The product law, also known as the "AND" or "BOTH" rule, helps us find the probability of two or more events occurring as long as they are independent of each other. This means that they can all happen at the same time. To use the product law, we multiply the probabilities of all individual events occurring.

So, if you see phrases containing the word "or", use the sum law. For example, "what is the probability of this OR that occurring?" And if you see phrases containing "and" or "both", use the product law. For example, "What is the probability of this AND that occurring?"

## Application of Probability in Genetics

Probabilities in genetics are even more useful with the Punnett square. This square is ideal for examining crosses of two alleles at one or two genes at most. For example, if you want to examine a cross between Aa and aa, you will get four boxes to examine the genotypes of offspring. This is because there are two possible alleles for one gene (see Fig. 1). The Punnett square is a powerful tool for predicting the probability of offspring inheriting specific traits from their parents.

when examining a cross like AaBb x AaBb, you 16 boxes to examine the genotypes of offspring (see Fig. 2). This is because there are two possible alleles for each of the two genes being examined. As you can see, the number of squares required increases exponentially with each additional gene being examined. For a three-gene cross, the number of squares required would be even higher. It's easy to see how meticulous the work can become when using a Punnett square of that size.

Ultimately, beyond a certain point, Punnett squares are not a feasible option. Therefore, probability and simple mathematics are required. Even for simple, single-gene crosses, probability can be used to double-check our Punnett squares.

### Examples of Probability in Genetics

When breeding two homozygous plants with different traits, such as tall (TT) and short (ss), we can assume Mendelian inheritance principles. This that there is complete dominance, independent assortment, and segregation of alleles. Since the tall trait is dominant over the short trait, we know that each parent must have at least one T allele in their gametes. Since both parents are homozygous, they only have one type of allele to give. Therefore, the tall parent will give a T allele, and the short parent will give an s allele to each of their offspring. This results in the first filial generation having only one genotype, which is Ts (see Fig 3).

Now let us continue by crossing two F1 plants together. This kind of cross (F1 x F1) is a monohybrid cross because both plants are hybrid (heterozygous) for the same gene. Without doing a Punnett square, we know that each parent can give either T or s alleles, and those two alleles can combine with the other parent's allele (Fig. 4).

Ts and sT are the same; thus, there are three possible genotypes: TT, Ts, ss. We know the proportions they occur in as well because, for this cross, we get the TT genotype once, the ss genotype once, and the Ts genotype twice.

So the probability of getting a homozygous tall plant: Pr (TT) = 1/4. The probability of getting a heterozygous tall plant: Pr (Tt) = 2/4 = 1/2. The probability of getting a homozygous short plant: Pr (tt) = 1/4.

The two pure-breeding parents are called P generation, and when crossed, their descendants are F1. If the F1 x F1 cross is performed, their descendants are F2. If you cross F2 x F2, their descendants are F3, and so on.

Figure 5: F1 x F1 = F2. Bank of biology.

We discovered these probabilities through reasoning (Fig. 5). Now, let's use the sum and product rule to answer further questions. It's important to analyze each part of each question before combining them for your final answer.

What is the probability of having a tall plant in the F2 generation? Tall plants can have TT or Ts genotypes. So we are looking for Pr (TT or Ts).We remember that OR in probability signifies addition. Therefore, according to the sum rule:Pr (TT or Ts) =  Pr (TT) + Pr (Ts) =

What is the probability of having two tall offspring in the F2 generation? Each offspring has a 3/4 chance to be tall. We discovered this in the previous question. But, we want two tall offspring this time. We need offspring #1 AND offspring #2 to be tall. This is Pr (TT or Ts) AND Pr (TT or Ts)Therefore, according to the product rule: Pr (tall) x Pr (tall) =

What is the probability of having one short and one tall offspring in the F2 generation? We know each offspring has Pr (tall) = 3/4.Because there are only two possible phenotypes for this trait, to find the probability of short offspring, we simply do:Pr (short) = 1 - Pr (tall) = We need to find Pr (short) AND Pr (tall), which means multiplication. Therefore according to the product rule: Pr (short) x Pr (tall) =

Now, let's look at a more challenging example—the dihybrid cross.

Dihybrids are organisms that are heterozygotes at two different alleles.

Let's use human genetics and traits for this example to demonstrate that this kind of probability analysis is possible in higher organisms. Many genes in human beings do not follow the principles of Mendelian genetics, but here are two that do. The alleles for freckles and for a widow's peak (that unique V-shaped hairline) are dominant at their respective gene loci. Therefore, the alleles for no freckles and no widow's peak (a more rounded or straight hairline) are recessive.

What would happen if two people who are dihybrids for widow's peaks (Ww) and freckles(Ff) traits married and had children? What kind of alleles could they make? What is the possible genetic outcome for their offspring? Let's examine and answer such questions.

Q: What do dihybrids for widow's peaks and freckles look like?

A: They would both have freckles and widow's peaks because they are heterozygotes for those traits, which are dominant.

Q: What are the genotypes for these parents?

A: FfWw for both parents.

Q: Since both parents have the same genotype (for these two traits), they can make the same possible alleles in their gametes. What are these possibilities?

A: Each gamete must contain one allele of each gene; therefore, FW, Fw, fW, and fw are the possible gametes.

Q: What are the possible genetic outcomes for their offspring?

A: Let's use a Punnett square to examine this. For a dihybrid cross, Punnett squares are quite large and unwieldy (16 squares), but we will simplify this with probabilities later.

To summarize, dihybrid crosses involve two heterozygous organisms and give a phenotypic ratio of 9:3:3:1. We can use probability laws to predict the outcomes of genetic crosses, even without using a Punnett square. For example, to find the probability of having two children with freckles and widow's peaks, we need to find the probability of having a single child with both traits and then multiply that by itself for two children. Using the sum and product law, we can find that the probability of having a single child with freckles and widow's peaks is 9/16, and the probability of having two children with both traits is less than 1/3. This demonstrates the power of probability analysis in genetics and shows how it can help us better understand the possible outcomes of genetic crosses.

Probabilities in Genetics - Key Takeaways There are two key laws of probability in genetics: the sum law and the product law. The sum law is also called the OR law, meaning you use it to add two probabilities together when determining probabilities of one OR the other possibility. The product law is also called the AND law, meaning you use it to multiply probabilities with each other when determining the probabilities of one AND another possibility. Punnett squares are best used for single-gene analysis of traits that obey Mendelian genetics, If more complex genetic assessments are required, probabilities and simple mathematics are necessary to help analyze offspring potential genotypes and phenotypes.

## Probabilities in Genetics

What is probability in genetics?

Probability in genetics are simple mathematical analysis of proportions to help understand and postulate future inheritance patterns.

How is probability used in the study of genetics?

Probability is used in genetics to help analyze possible offspring genetic outcomes.

How can the principles of probability be used in genetics?

The principles of probability, namely the sum law and the product law, can help to calculate the likelihood of a genotype or phenotype in offspring of a cross.

How is probability important in genetics?

Probability is important in genetics because it helps us determine genotypic and phenotypic outcomes for future offspring.

How is probability used to predict traits?

Probability helps predict traits by making a mathematical calculation of the likelihood of a traits existence in a given offspring, via the sum law and the product law.     