I saw this episode of "What Would You Do?" a few months ago, and I keep wondering what would statistically be the best thing to do in this situation.
Here is the problem formulation:
You are waiting in line at a supermarket, where there is a sign saying that the millionth customer will win an amazing prize. Just before your turn, someone in a hurry asks you if he can cut in line in front of you.
Question: If you let this person cut in front of you, does he have a higher probability of winning the prize than you?
This leads to a more general question which is for me a paradox:
If you are in line and want to be the millionth customer, should you try to go as soon as possible, or should you wait? On one hand the more you wait (the more you let people cut in front of you), the closer you get to the millionth. On the other hand the more you wait, the more you risk of someone being the millionth before you.
We can formalize it as follows:
You are customer number $n$. Of course you don't know your number $n$, but you know that you will win if $n=N$ with $N=1000000$. The only thing you know is that $n<=N$ (no one as won the prize yet, but you might). Being the $nth$ customer, your probability to win is then: $\text{Prob}(n=N|n<=N)$. But if you let someone cut in front of you, you become the $(n+1)$th customer and your probability to win becomes $\text{Prob}(n+1=N|n<=N)$.
Now here is my question:
Is there a way to compare $\text{Prob}(n=N|n<=N)$ and $\text{Prob}(n+1=N|n<=N)$?
As with most probability questions drawn from the real world, this one is impossible to answer without having in mind a specific hypothesized distribution on the number of customers already served (call that number $S$).
We might speculate, based on the best information we have about the supermarket, that $S$ is drawn from a normal distribution with mean $\mu$ and some standard deviation (rounded to the nearest integer, and truncated to take into account $S<N$). In that case, it can presumably be worked out that we should wait about $N-\mu$ customers before taking our turn.
One example that illustrates how dependent this question is on the hypothesized distribution of $S$: if you knew for a fact whether $S$ was even or odd, that would completely influence whether you kept your spot or let one customer pass you.