Suppose there's a lottery. Each ticket sold has probability $p$ of winning, and they are all independent of each other. The size of the jackpot is $j$. If there are $n$ winners, each winner gets a payoff of $j/n$ dollars. The total number of tickets sold is $t$.
What is the expected value of a lottery ticket? Also, given that I win, what is the probability that I have to share the jackpot with at least one other person?
PS – I think I know the answer, but have failed to convince someone else, so I'm looking for a third-party to give an answer.
Best Answer
Probability that no one wins: $(1-p)^t$.
Expected value for a particular ticket: $ \dfrac j t (1-(1-p)^t)$. (Pot times probability that the pot is distributed.)
Probability that you win alone: $p(1-p)^{t-1}$
Probability of shared win: $p-p(1-p)^{t-1}$
Probability of sharing given that you win: $\dfrac{\text{prob of shared win}}{\text{prob win}}=\dfrac{p-p(1-p)^{t-1}}p = 1-(1-p)^{t-1}$
This is just one minus the probability of everyone else losing because of independence.