Question

Probabilities are sensitive to the form of the question that was used to generate the answer

(Source: Minka.) My neighbor has two children. Assuming that the gender of a child is like a coin flip, it is most likely, a priori, that my neighbor has one boy and one girl, with probability 1/2. The other possibilities—two boys or two girls—have probabilities 1/4 and 1/4.

a. Suppose I ask him whether he has any boys, and he says yes. What is the probability that one child is a girl?

b. Suppose instead that I happen to see one of his children run by, and it is a boy. What is the probability that the other child is a girl?

Answer 1

Answer:-

Given That:-

My neighbor has two children. Assuming that the gender of a child is like a coin flip, it is most likely, a priori, that my neighbor has one boy and one girl, with probability 1/2. The other possibilities—two boys or two girls—have probabilities 1/4 and 1/4.

a. Suppose I ask him whether he has any boys, and he says yes. What is the probability that one child is a girl?

a) P(one child is girl |at least one boy) =P(both girl and boy) /P(At least one boy) =(1/2)/(3/4) =2/3

b. Suppose instead that I happen to see one of his children run by, and it is a boy. What is the probability that the other child is a girl?

b) Due to independence:

P(other is a girl| given 1 is boy) =P(other is girl)=1/2