There are 10 tickets in a lottery. I have two of them. The prizes are 10,5,2 dollars . Two of my friends X and Y come to me and say that I won.
Probability that X is saying true is 3/4 and Y if saying true is 1/3.
What is my expectation. (I would love a detailed explanation on dealing with my friends)
Attempt: My probability of winning is 1/5 so if my friends don't say anything then my expectation is 17/5 . I thought that X is more trustable so I can rely totally on him but I am confused there.
your chances of having won are thus 5/6, because the probability that both are liars is $\frac{1}{4}\frac{2}{3} = \frac{1}{6}$, so there is a $\frac{5}{6}$ chance that you have really won.
we now know that at least one of your tickets is a winner; the expectation from the first is 17/3. The second may have won, with probability 1/9. It's expectation is again 17/3(I think) so your total expectation is
$\frac{5}{6}\frac{17}{3} + \frac{5}{6}\frac{1}{9}\frac{17}{3}$