Suppose two people have a yes / no question that the probability for each of them to answer correctly is $\ p $. Each one answer is independent of the other one. What way is the best for them to pick an answer?
- Picking one of their answer randomly
- If they both think the same answer they will say it if they each choose different one they will pick randomly one of their answer and say it.
If I set $\ A_i $ = probability of person $\ i = 1,2 $ answer correctly. $\ B $ will be the event of them picking both the same answer. $$\ P(B) = ((A_1 \cap A_2)\cup (A_1^c\cap A_2^c))$$ if they just pick randomly is
$$\ P(A_i) =p\\ P(A_1 \cup A_2) = P(A_1) + P(A_2) - P(A_1 \cap A_2) = 2p -p^2 $$
According to the second way, if they both pick the same answer :
$$\ P( right \ answer | \ B) =\frac{P((A_1\cap A_2)\cap B)}{P(B)} = \frac{p^2}{p^2+(1-p)^2} $$
and then their last option is
$$\ P( right\ answer | \ B^c ) = \frac{P((A_1\cup A_2)\cap B^c)}{P(B^c)} = \frac{P((A_1\cup A_2)\cap((A_1\cap A_2^c)\cup(A_1^c \cap A_2)))}{P((A_1\cap A_2^c)\cup(A_1^c\cap A_2))} = \frac{P((A_1\cap A_2^c)\cup(A_1^c\cap A_2))}{P((A_1\cap A_2^c)\cup(A_1^c\cap A_2))} = 1 $$
Which is wrong.. Yet I couldn't find what am I missing?