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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)? (b) Suppose that when the black rat is mated with a brown rat, all 5 of their offspring are black. Now, what is the probability that the rat is a pure black rat?

Short Answer

Expert verified
(a) The probability that the black rat is pure black (BB) is 25%. (b) Based on the new information that all offspring of the black rat and the brown rat are black, the probability that the rat is a pure black rat (BB) is 100%.

Step by step solution

01

Determine the genotype of the parents given the information about the siblings

Because black dominates brown, rats can have either of the three genotypes: - BB (pure black, with both black genes) - Bb (hybrid, with one black gene and one brown gene) - bb (brown, with both brown genes) Since the black rat in question has a brown sibling, both parents cannot be BB (pure black), because that would result in only BB offspring. Therefore, we can conclude that both parents are Bb (hybrids).
02

Calculate the probability of the black rat having a pure black genotype (BB)

Since both parents are Bb, we can create a Punnett square to find the probability of each of the offspring's genotypes: \[ \begin{array}{c|c|c} & B & b \\ \hline B & BB & Bb \\ \hline b & Bb & bb \end{array} \] Looking at the Punnett square, there is a 1/4 chance of having a BB offspring, a 1/4 chance of having a bb offspring, and a 2/4 (1/2) chance of having a Bb offspring. Thus, the probability of the rat in question being a pure black rat (BB) is \( \frac{1}{4} \), or 25%. (a) Therefore, the probability that this rat is pure black is 25%.
03

Calculate the offspring's genotypes when the black rat mates with a brown rat

Now when the black rat (B? genotype) mates with a brown rat (bb genotype), all 5 of their offspring are black. This update leads us to evaluate the probability of the black rat being BB again. Let's consider the two situations when the black rat is either pure black (BB) or hybrid (Bb): 1. B? = BB (pure black rat) If the black rat is BB, and it mates with a bb rat, all offspring will be Bb (black). 2. B? = Bb (hybrid black rat) If the black rat is Bb, and it mates with a bb rat, the offspring can be either Bb or bb. Since we are given the information that all offspring are black, the pure black (BB) genotype aligns with this information.
04

Update the probability based on the new information

In light of the new information that all offspring of the black rat and the brown rat are black, it becomes clear that our black rat is most likely a pure black (BB) rat. Since all offspring of a BB x bb mating are black, we can say that this event supports the hypothesis that the black rat is pure black (BB). (b) Now, since we know all offspring turned out black, the probability that the rat is a pure black rat (BB) is 1, or 100%.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Punnett Square
The Punnett Square is a vital tool in genetics for predicting the probability of an offspring having particular genotypes, based on the genetic makeup of the parents. It is essentially a grid that helps visualize the potential combinations of parental genes during reproduction. Each box in the Punnett Square represents a possible genotype of the offspring.

To illustrate this, let's consider a scenario where both rat parents have the genotype Bb, meaning they each carry one allele for black fur (B) and one for brown fur (b). Using a Punnett Square, you will place each parent's genetic contribution along the top and side of the grid:
  • The top of the grid shows the father's genotypes: B and b.
  • The side of the grid shows the mother's genotypes: B and b.
This setup allows us to fill out the grid, showing possible offspring genotypes: BB, Bb, and bb. From the square, you can visually calculate the probability:
  • 25% chance for BB (pure black)
  • 50% chance for Bb (hybrid black)
  • 25% chance for bb (brown)
This tool is fundamental for understanding how traits are inherited and can help predict the outcome of different genetic crosses.
Genotype
In genetics, a genotype refers to the specific genetic makeup of an organism, particularly in relation to a specific trait. For example, in this case of rat fur color, the genes come in pairs and include alleles B (for black) and b (for brown). These alleles form different genotypes such as BB, Bb, and bb.

Genotypes directly dictate the potential phenotype—the observable traits—of an organism:
  • BB: This genotype consists of two dominant black alleles, expressing the black fur phenotype. These rats are considered pure for the black trait.
  • Bb: This hybrid genotype includes one dominant black allele and one recessive brown allele, still expressing the black fur phenotype due to the dominance of the B allele.
  • bb: This genotype consists of two recessive brown alleles, and the phenotype is brown fur because there is no dominant allele to mask the recessive one.
Understanding genotypes is crucial because it provides insight into how traits are passed from one generation to the next, influencing the organism's characteristics based on the genetic code.
Dominant and Recessive Traits
Dominant and recessive traits are key concepts in genetics that explain how different traits are inherited through alleles.
  • Dominant Trait: This trait is expressed when at least one dominant allele ( B ) is present. In our rat example, black fur is a dominant trait. Even if the rat carries one allele for black and one for brown (Bb), the black-colored trait will be visible due to the presence of the dominant B allele.
  • Recessive Trait: This trait only appears when an organism has two recessive alleles ( bb ). In this case, a rat needs two brown alleles to have brown fur show up visibly, as the recessive trait is typically masked by the presence of a dominant allele.
The dominance and recessiveness of these traits influence genetic outcomes in populations and dictate which characteristics tend to recur and which are less commonly observed. Recognizing these patterns helps in predicting hereditary traits and understanding genetic diversity.

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Most popular questions from this chapter

Suppose that 5 percent of men and \(.25\) percent of women are color blind. A colorblind person is chosen at random. What is the probability of this person being mal?? Assume that there are an equal number of males and females. What if the population consisted of twice as many males as females?

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The Ballot Problem. In an election, candidate \(A\) receives \(n\) votes and candidate \(B\) receives \(m\) votes, where \(n>m\). Assuming that all of the \((n+m) ! / n ! m !\) orderings of the votes are equally likely, let \(P_{n, m}\) denote the probability that \(A\) is always ahead in the counting of the votes. (a) Compute \(P_{2,1}, P_{3,1}, P_{3,2}, P_{4,1}, P_{4,2}, P_{4,3}\). (b) Find \(P_{n, 1}, P_{n, 2}\) (c) Based on your results in parts (a) and (b), conjecture the value of \(P_{n, m}\). (d) Derive a recursion for \(P_{n, m}\) in terms of \(P_{n-1, m}\) and \(P_{n, m-1}\) by conditioning on who receives the last vote. (e) Use part (d) to verify your conjecture in part (c) by an induction proof on \(n+m\).

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