Ten million people enter a certain lottery. For each person, the chance of winning is one in ten million, independently.
Congratulations! You won the lottery. However, there may be other winners. Assume now that the number of winners other than you is W ⇠ Pois(1), and that if there is more than one winner, then the prize is awarded to one randomly chosen winner. Given this information, find the probability that you win the prize.
I cam up with the following: V-"stands for victory"
$P(V)=$$\sum _{n=0}^{\infty}P(V|W)P(W)$
Can anybody give me a hint of how to find $P(V|W)$?
HINT: There may be a better way of going about this.
Let $A$ be the event that you have won, $P(A) = 1/N = 10^{-7}$, where $N$ is the number of people. Let $B$ be the event that at least one person won, $P(B) \approx 1-e^{-1}$ (from the Poisson).
Since everyone's chances of getting the prize are the same, $P(V) = P(B)/N$. What you actually wanted, it seems, is "victory" (getting prize) conditional on "having won" which is $$ P(V|A) = {P(V\cap A)\over P(A)} = $$ [I'll let you finish it.]