TITLE: Husband and wife independently give correct answers with probability p
QUESTION [2 upvotes]: A true-false question is to be posed to a husband-and-wife team on a quiz show. Both the husband and wife will independently give the correct answer with probability $p > \frac12$.
Which of the following is a better strategy for the couple? 
(a) Choose one of them and let that person answer the question.
(b) Have them both consider the question, and then either give the common answer if they
agree or, if they disagree, flip a coin to determine which answer to give.
What if $p < \frac12$? What if $p=\frac12$?
All I think I understand is that if $p=\frac12$ then (a) and (b) are the same since the probability of the wife or husband getting the answer correct is independent from each other.
I'm not really sure how I am supposed to do this. I would just like some hints on how to tackle this problem. I'm really struggling in my probability class, so if anyone knows of any useful resources I would love to hear about them.

REPLY [4 votes]: If the husband and wife are both correct with probability $p$ then there is $p^2$ chance they will agree on the correct answer, $(1-p)^2$ chance they will agree on the wrong answer, and $2p(1-p)$ chance they will disagree, which consists of $p(1-p)$ for the husband is right and wife wrong, and $p(1-p)$ for the wife is right and husband wrong.
If you throw a fair coin to decide what to choose when they disagree, the total chance of them being correct is thus $p^2+1/2\times2p(1-p)=p$ which is the same probability as either one of them being right, so you may as well just pick one of them. This is completely unrelated to the value of $p$.
If $p=1/2$ then you could just as well throw a coin. If $p<1/2$ then the best choice is the opposite of whatever they guess is correct.
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