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?

Answer:a) (1/2)^3 / (x^3 +  (1/2)^3)   b) (1/2) (1/2 + 1 - x)Where x is the probability that a prince has not hemophilia given that the queen is not a carrierStep-by-step explanation:We can use Bayes' theorem to calculate this porbability, let:A = The event for the queen has hemophiliaB = The event for a prince to have hemophiliaWe are looing for the porbability P(A | 3B)Using Bayes'  theorem:P(A | 3B) = P(3B | A) ( P(A) / P(3B) )We know that:P(3B | A) = (1/2)^3P(A) = 1/2Now let's calculate P(3B), here we will assume that the porpability that a prince does not have hemophilia given that the queen is not a carrier is xThereforeP(3B) = (1/2)x^3 + (1/2) (1/2)^3Replacing all the values:P(A | 3B) = (1/2)^3 / (x^3 +  (1/2)^3)If x=1, that is, if it is 100% probable that the prince will not have hemphilia given that the queen is not a carrier:P(A | 3B) = 0.111If x=0.9, that is, if it is 90% probable that the prince will not have hemphilia given that the queen is not a carrier:P(A | 3B) = 0.121Now lets calculate the probabiity for a fourth prince to have hemophilia:If the queen has hemophilia, the probability is(1/2)If the queen does not have hemophilia, the probability is:1-xTherefore, the total probabilty is : (1/2) (1/2 + 1 - x)If x = 1, the probability is 1/4If x=0.9, the porbaility is 0.3