I'm stuck on a probability question. I have a partial solution to the answer, but I can't get further.
The problem is as follows: consider a wine tasting experiment, we will ask our participant Bob to taste $N$ glasses of wine. Each glass is filled with either a cheap table wine or a grand cru, suppose there are exactly $k$ glasses of cheap wine.
Our participant Bob is a novice to wine tasting, so can not distinguish between the two types of drinks. We let Bob drink each glass and ask him whether it is a cheap wine or a grand cru. What is the probability of Bob correctly distinguishing $S$ cheap table wines, given that he does not know that there are k of them?
I think the solution to this problem is as follows: considering Bob does not know how many glasses are table wine and can't distinguish between the 2 types, the amount of times $L$ of him identifying a glass as table wine is binomially distributed with parameters $N, 0.5$. Now I suspect that the amount of correctly identified glasses of table wine $S$ is also binomially distributed with parameters $L, 0.5$. Because again, Bob can not distinguish between either wine. Is this correct?
We make the (only possible) assumption w/o any other data, viz. the probability of any glass being correctly identified is $0.5$, and we are asked to find out Pr($s$ table wines are correctly identified)
Now it doesn't matter which glasses contain the table wine; for each glass of wine tasted (table or grand cru), the probability that he is correct will be 0.5
So it should be clear that the pmf for table wine correctly identified will be $X\sim binom(k,0.5)$,
and although not asked for, that of grand cru will be $Y\sim binom(n-k,0.5)$
and pmf of total successes will be $Z\sim binom(n,0.5)$
So finally, $P(X=s) = \binom{k}{s}\cdot 0.5^s\cdot0.5^{k-s}$