Solving a Genetics Problem Involving Rat Coat Colors

 Genetics is the study of how traits are passed down from one generation to the next. One example of genetics is the coat color of rats. In this blog post, we will solve a genetics problem involving rat coat colors. We will use Punnett squares and probability to find the probability that a rat with two black parents and a brown sibling is a pure black rat.

Problem:

In a certain species of rats, black dominates over brown. Suppose that a black rat with two black parents has a brown sibling.

(a) What is the probability that this rat is a pure black rat (as opposed to being a hybrid with one black and one brown gene)?

Solution:

Let's begin by defining some terms that will help us solve the problem.

Black is the dominant trait, and brown is the recessive trait. The allele for black is denoted by B, and the allele for brown is denoted by b.

A pure black rat has two black alleles, represented as BB. A hybrid rat has one black allele and one brown allele, represented as Bb.

Now, let's create a Punnett square to represent the possible offspring of two black rats:

  | B  | B

--|----|---

B | BB | BB

--|----|---

B | BB | BB

The Punnett square shows that if two black rats mate, their offspring will be pure black with a probability of 1.

However, we are given that one of the offspring is brown. This means that the two black rats are both Bb. Let's create a new Punnett square to represent the possible offspring of two hybrid rats:

  | B  | b

--|----|---

B | BB | Bb

--|----|---

b | Bb | bb

The Punnett square shows that if two hybrid rats mate, their offspring will be pure black with a probability of 3/4 and hybrid with a probability of 1/4.

Given that the black rat has two black parents and a brown sibling, the most likely scenario is that the black rat's parents are both Bb. This means that the black rat is either pure black (BB) or a hybrid (Bb).

Let P(BB) be the probability that the black rat is pure black, and P(Bb) be the probability that the black rat is a hybrid. The probability of the black rat having two black parents who are both Bb is:

P(Bb, Bb) = 1/4

The probability of the black rat being a pure black rat given that its parents are both Bb can be calculated using the conditional probability formula:

P(BB | Bb, brown sibling) = P(BB and Bb, brown sibling) / P(Bb, brown sibling)

We know that the brown sibling is a bb genotype. Thus:

P(BB and Bb, brown sibling) = 0

P(Bb, brown sibling) = 1/2

Substituting the values, we get:

P(BB | Bb, brown sibling) = 0 / (1/2) = 0

Therefore, the probability that the black rat with two black parents and a brown sibling is a pure black rat is 0. The black rat is most likely a hybrid (Bb).

We solved a genetics problem involving rat coat colors. We used Punnett squares and probability to find the probability that a black rat with two black parents and a brown sibling is a pure black.