I have two children. One of them is a boy born on a Tuesday. What is the probability they are both boys?
2 answers
I think it's 1/2, since you already know one is a boy for sure.
First, your reasoning is wrong because you don't know which child - or even if both - fit the description. But trying to explain why that matters makes many people ignore other considerations.
As the famous Martin Gardner observed in 1959 about this question's predecessor, it is ambiguous. I have to know why you choose to tell me this very odd piece of information.
If I walk up to people at random on the street, and ask "Pardon me, but do you happen to have exactly two children, including at least one boy who was born on a Tuesday?" Then the answer is 13/27. There are 14*14=196 possible combinations of Gender+Day, but only 27 of them include a Tuesday Boy. Of those 27 combinations, 13 are two boys.
But if you volunteered the information, I have to assume that you randomly choose one of the one (possible but unlikely) or two (most likely) descriptions that applied to your children. In that case, I have to consider the possibility that you would have said "One of them is a girl born on a Monday" even though her brother was born in Tuesday. So of the 27 combinations with a Tuesday Boy, you would tell me that fact only 14 times. Of those 14, 7 include two boys, and the answer is 1/2.
As the famous Martin Gardner observed in 1959 about this question's predecessor, it is ambiguous. I have to know why you choose to tell me this very odd piece of information.
If I walk up to people at random on the street, and ask "Pardon me, but do you happen to have exactly two children, including at least one boy who was born on a Tuesday?" Then the answer is 13/27. There are 14*14=196 possible combinations of Gender+Day, but only 27 of them include a Tuesday Boy. Of those 27 combinations, 13 are two boys.
But if you volunteered the information, I have to assume that you randomly choose one of the one (possible but unlikely) or two (most likely) descriptions that applied to your children. In that case, I have to consider the possibility that you would have said "One of them is a girl born on a Monday" even though her brother was born in Tuesday. So of the 27 combinations with a Tuesday Boy, you would tell me that fact only 14 times. Of those 14, 7 include two boys, and the answer is 1/2.
1 answer