Ok, I just need help showing the answer to this problem mathematically.

A child has a 1/2 probability of inheriting a certain disease. In a two child family, what is the probability that EXACTLY one child will inherit the disease?

Obviously, there are four total outcomes.

{YY, YN, NY, NN}

And from this, I can see that there are two instances in which exactly one child inherits the disease, so the answer is 1/2.

I just need help with figuring out how to write this out mathematically. I know I'm just missing something simple.

I thought maybe it was just:

A= child 1 is affected, B = child 2 is affected

Pr(AUB)

But

Pr(AUB) = Pr(A)+P(B) - Pr(AnB) = 1/2 +1/2 -1/4 = 3/4

Which makes sense because the probability of A or B is:

The probability that child 1 is affected + the probability that child 2 is affected + the probability that both are affected which totals 3/4

Is the proper way to mathematically write out what I'm looking for:

Pr(AUB) - Pr(AnB) = [Pr(A)+P(B)-Pr(AnB)] - Pr(AnB)

= (1/2 + 1/2 - 1/4) - 1/4 = 1/2

Doing it this way makes sense to me since I'm looking the probability of A or B but not Both, but it seems funky to have to subtract Pr(AnB) twice.

Any push in the right direction is appreciated. Thanks