Recently, a good friend gifted me a bag of Easter SweeTarts, with ducks and bunnies in place of the usual round candies. They come in five flavors, which are officially blue raspberry (blue), cherry (pink), grape (purple), orange (orange) and green apple (green). (This according to Wikipedia, at least.)
I don't like the blue ones, so because I'm that way, I culled those out and they're still sitting in front of me in a bowl. The rest I ate at my leisure—though rather less leisurely than is quite becoming, I'm afraid.
At some point, near the end of the bag, I wondered what the probability was, if I pulled out two candies at random, that they'd have the same flavor. I looked at the candies that were in there, and it was a simple matter to figure the answer out. It was—ahh, but then I'd be giving the puzzle away.
I then turned to look away and puckishly reached into the bag and pulled out a pair of green apples. (Those are my second favorite, after the artificial grape ones, which, as you'll know if you eat them, taste nothing at all like actual grapes.) I wondered if that affected the probability at all—figuring that if anything, it would have reduced the odds—but after looking in and doing some quick figuring, I found that the odds were exactly the same as before.
"Hunh!" I thought to myself in surprise, and (if you know me at all, you know what happened next) I wondered what the odds were of that. As it so happens, that's not a question you can answer rigorously at all, without knowing the priors. But perhaps you can rigorously answer
Q1: What were the odds of drawing a matching pair of candies?
I reached in again, and this time pulled out a cherry and a grape; I did it again, and pulled out a matching pair of grapes. So this time, I thought, for sure there must be a change in the odds of drawing a match, but when I looked in again, the odds were once more exactly the same as the first time. What's more, I reached in again, pulled out a pair of green apples, looked in one last time, and the odds were again exactly the same as before!
Q2: Knowing there were no blue raspberry candies at all, how many of each of the other flavors were there, before I pulled out the first pair of green apples?