There is a raffle. Actually, 2 raffles. The first raffle is for prize “A”. The second raffle is for "prize “B”. Tickets sold separately. I bought a ticket for prize “A” only.
I get a call saying I had won prize “B”. When I asked about prize “A”, I was told my ticket must have inadvertently been left in the ticket tumbler and was drawn as the winner of the prize “B” raffle. Now, let’s say there were 200 tickets, I realize the winners odds are 1 in 200 but… my question is: Does the fact that the one ticket that was not supposed to be in the tumbler was the one that was drawn as the winner have any affect at all on the probability? Thanks.
More tickets → smaller probability. 201 instead of 200 tickets reduces the probability to 1/201 instead of 1/200.
You’re also assuming that only one ticket was left in the ticket tumbler.
And as a pure probability question, it’s easy: You would assume that any ticket in the tumbler has the same chance. But the real world often isn’t that simple. For instance, a ticket left in tumbler might have been left in because it got caught in an internal seam in the tumbler. If that’s the case, then it won’t get drawn unless it manages to first get itself unstuck, and so that ticket would have a lower chance than the others.
You could probably also come up with mechanisms by which that ticket might have a higher chance than the others. Which real-world phenomenon is the dominant one? Hard to say.
Thank you. In my feeble mind, I was assuming the odds of the one ticket that was not supposed to be there being drawn as the winner would have to have to be higher…but I guess it still all comes down to the odds being 1:# of tickets.
There’s also the question of whether the stuck ticket has increased your chance at winning. Intuitively, it seems like it should, since you’ve effectively entered in two raffles. But if we take the stuck ticket as a given, it means you couldn’t possibly have won the first raffle, and therefore your only chance was in the second–which is now 1/201.
If our prior is that some ticket got stuck, then the chance at you winning (assuming both raffles had 200 tickets) is 1/200 + 1/199*1/201 =~ 1/199. So the odds have improved slightly, but not a lot.