Problem 70 - A First Course in Probability

Question

There is a \(50-50\) chance that the queen carries the gene for hemophilia. If she is a carrier, then each prince has a \(50-50\) chance of having hemophilia. If the queen has had three princes without the disease, what is the probability that the queen is a carrier? If there is a fourth prince, what is the probability that he will have hemophilia?

Step-by-Step Solution

Answer

The probability that the queen is a carrier given that her three sons do not have hemophilia is approximately 11.11%. If there is a fourth prince, the probability that he will have hemophilia is around 5.56%.

Step 1: Understand the probabilities

We know that the queen has a 50% chance of being a carrier (C) and a 50% chance of not being a carrier (NC). If the queen is a carrier (C), each prince has a 50% chance of having hemophilia (H) and a 50% chance of not having hemophilia (NH). We are given that the first three princes do not have hemophilia (NH1, NH2, NH3).

Step 2: Apply Bayes' theorem

Bayes' theorem states that P(A|B) = (P(B|A) * P(A)) / P(B), where A and B are events. In this case, we want to find the probability that the queen is a carrier (C) given that her three sons do not have hemophilia (NH1, NH2, NH3). Therefore, we need to find P(C|NH1, NH2, NH3). P(C|NH1, NH2, NH3) = (P(NH1, NH2, NH3|C) * P(C)) / P(NH1, NH2, NH3)

Step 3: Calculate the probabilities

First, we find the probabilities: P(C) = 0.5 (given) P(NC) = 0.5 (given) P(NH1, NH2, NH3|C) = 0.5 * 0.5 * 0.5 = 0.125 (since the probabilities are all independent) P(NH1, NH2, NH3|NC) = 1 * 1 * 1 = 1 (since the probabilities are all 100% if the queen is not a carrier) Now, calculate P(NH1, NH2, NH3) using the law of total probability: P(NH1, NH2, NH3) = P(NH1, NH2, NH3|C) * P(C) + P(NH1, NH2, NH3|NC) * P(NC) P(NH1, NH2, NH3) = (0.125 * 0.5) + (1 * 0.5) = 0.0625 + 0.5 = 0.5625

Step 4: Plug the probabilities into Bayes' theorem

Now, we plug these probabilities into Bayes' theorem: P(C|NH1, NH2, NH3) = (0.125 * 0.5) / 0.5625 = 0.0625 / 0.5625 ≈ 0.1111

Step 5: Find the probability of the fourth prince having hemophilia

If the queen is a carrier (now with a probability of approximately 0.1111), there is a 50% chance that the fourth prince will have hemophilia (H4). So, the probability is: P(H4|C) = 0.5 We use the law of total probability again to find the probability that the fourth prince will have hemophilia (H4), given the new information that the queen is likely a carrier: P(H4) = P(H4|C) * P(C|NH1, NH2, NH3) + P(H4|NC) * P(NC|NH1, NH2, NH3) P(H4) = (0.5 * 0.1111) + (0 * (1 - 0.1111)) ≈ 0.0556 So, the probability that the fourth prince will have hemophilia is around 5.56%.