For any individual x born in Transylvania with a vampire father, there is a 50% chance that x is a vampire, independently for each birth. These are the only conditions under which a new vampire can be created. 75% of the Transylvanian males are vampires. Suppose Igor, a man who has lived in Transylvania his whole life, has three children that are not vampires.
(a) What is the probability that Igor is a vampire?
(b) If Igor has a fourth child, what is the probability that child will be a vampire?
For a, i have tried the set up of P(Vampire| Father is V) = .5
P(Not V| Father is V) = .5
P(V | father Not V) =0
P(Not V| Father not V) = 1
P(Father is V) = .75
I am just not sure what to do with the information that the 3 kids are non vampires. How would this fit into finding the probability that igor is a vampire?
Let's assume both a vampire and a non-vampire are equally likely to have 3 children. (I think that was a missing assumption).
Then, out of the population of fathers of 3,
25% are humans with 3 children, and 75% are vampires with 3 children. Out of the (father of 3) vampires, only 1/8 of them have 3 human children. So that would be $0.75/8$ = 9.375% of the 3-father population. On the other hand, all humans of the 3-father population have 3 human children. So, in total 9.375 + 25 =34.375% of the 3-father population have 3 human children. By ratio to those of them that are vampire, we get 9.375 / 34.375 = 0.(27) of the 3-father population with 3 human kids are vampires.
So, the probability Igor is a vampire is 27.(27) %.
Now, for the forth child, if Igor is human , it's 0% vampire, if Igor is vampire, it's 50%. So we get 27.(27) /2 = 13.(63)% is the probability the forth child will be vampire, namely half the probability Igor himself will be vampire.
Without any prior probability on the likelihood of having 3 children (like the one I started with, from above), you have no way of integrating that information in the probability. If humans are more likely to have children than vampires, that biases things more toward Igor being human. If the reverse is true, that biases things more toward him being vampire. The assumption I started with seemed the most reasonable/fair.