As I was driving back from the grocery store today, I saw a sign in a convenience store window that I had seen before.  It says “we sold a winning lottery ticket.”  I started thinking about whether that was a smart thing to advertise.  On the one hand, some people might foolishly think that it’s a “winning store” and buy from them thinking they’re lucky or something.  But on the other hand, others might avoid buying lottery tickets there because what are the odds that the same place will sell two winning tickets?  And yet, it shouldn’t matter either way, because the probability that you will buy a winning ticket is completely independent of where you buy it, or what that store’s history of ticket sales is.

So here’s what I’m confused about:  On the one hand, from a buyer’s point of view, whether a store has ever sold a winning ticket before shouldn’t have any bearing on their choice to buy there.  Their odds of winning are exactly the same no matter where they go.  Yet from the store’s perspective, the chance that they will sell two winning tickets in a given amount of time is far lower than the chance that they will sell one during that period.

If it’s so much less likely that a store will sell two winning tickets than one, and yet no less likely that a customer would buy a winning ticket there than any other place in a given lottery, what should a buyer conclude about whether they should buy there?  I’ll admit, I didn’t pay much attention in statistics and may be misunderstanding something, but could anyone explain this apparent paradox?  Thanks!

Update:

After getting some responses from a few people, I realized I may need to clarify my question, so maybe this will make things clearer:

I guess my question is more general than I made it seem, and not really about the psychological effect of the store’s sign either. I’m really just interested in how you reconcile the two probabilities (and lotto ticket may not be the best example since it introduces the variable of the buyer getting to choose their number.)

I’ll simplify the problem. Let’s say there’s a sweepstakes where there is exactly one winner per contest, and the selection is made by random selection from among the many distributed tickets (customers don’t choose their ticket/number, either, to simplify things). The sweepstakes runs several times and there is a large number of stores distributing tickets.

The probability that the same store will sell at least two winning tickets is much lower than the probability that they will sell at least one, but the probablity of receiving a winning ticket is the same for every ticket, independent of where it is sold.

So here’s the question. By getting your ticket at a store that has previously distributed a winning ticket, aren’t you in effect betting that not only will you win (the odds of which are the same no matter where you go), but also that the store you’re at will distribute at least two winning tickets (the odds of which are lower than distributing at least one)? This reasoning suggests that it would be better to get your ticket at a store that had never distributed a winning ticket. Yet, the odds of a particular ticket winning is independent of the store’s past history. I know technically there’s no reason not to get your ticket at the place that had previously distributed a winning ticket, but how do you get around the fact that you’re essentially also betting on that store giving two winning tickets (whereas by going somewhere else, you’re only betting that they will give at least one)?