effect of knowing birthday of one person and the gender on probability of the second gender

Question

Consider there are two applicants for a job. One of the applicants is a man, and it is known that he was born on a Wednesday. What is the probability that the second applicant is also a man? (I am sure its not 1/2)

Answer 1

This is an exceedingly ambiguous question (not your fault).

If we phrase the question:

1) We have two applications with a combination of birthdays and gender; we know that one is a male, born on Wednesday; what is the probability both applicants are men.

Then the answer is that of the $14^2$ possible combinations $27$ of them involve a man born on a wednesday. (14 ways the first applicant is a Man born on Wednesday and the second could be any combination, plus 14 ways the second is a Man born on a wednesday and the first application could be any combination, minus one case of "double counting" when both applicants are men born on wednesday). Of those 27 options 14 involve a woman $(MW, WS-Sa),(WS-Sa, MW)$ and 13 involve two men. So the probability is $\frac {13}{27}$ slightly less than $\frac 12$.

But suppose we phrase a different question:

2) We have two applicants. We pick up one applicant and see that it is a Man born on a Wednesday. What is the probability the other is a Man.

Well.... $\frac 12$ duh... The first application has no bearing on the second.

.....

So which version is correct? In most mathematical problems where we say something like "We flip a coin two times and one time was a head, what's the probability the other is a head" we interpret it the first way. But I think linguistically that "one is a male born on a Wednesday" is so specific it can clearly be interpreted as "Of the two applicants, George Porgieboy, is a man born on a wednesday; what's the probability that the other applicant, E. Mendoza, is a man".

In fact I'd say "We flip a coin two times and one time was a head, what's the probability the other is a head" is badly stated. We can't have "THE other" if we aren't saying "THIS one is a head".

I'm going to leave the interpretation to you. But as a mathematical question it is paramount that we understand questions of the type of type 1. However I'd say it is the responsibility of the person asking the question to be precise and clear as to what the interpretation is.

Those who insist it MUST be 1) I think are bullies and incorrect. I think linguistically nearly everyone in would interpret it as 2) "One of the applications, George Porgieboy, is a boy born on a Wednesday".