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?

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 carrier

Step-by-step explanation:

We can use Bayes' theorem to calculate this porbability,

let:

A = The event for the queen has hemophilia

B = The event for a prince to have hemophilia

We 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) ^3

P (A) = 1/2

Now 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 x

Therefore

P (3B) = (1/2) x^3 + (1/2) (1/2) ^3

Replacing 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) = 0.111

If 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) = 0.121

Now 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-x

Therefore, the total probabilty is:

(1/2) (1/2 + 1 - x)

If x = 1, the probability is 1/4

If x=0.9, the porbaility is 0.3