I was trying to solve this problem and stumbled on a subproblem. What is the probability that one person does NOT see their name when it is in one of 100 identical boxes and they can open 50 of them?
My gut told me there is a 50% chance I don’t see my name, but another part of me said, there is a 99/100 chance I don’t see my name on the first try, a 98/99 chance I don’t see it on the second try, etc. Should I add or multiply all those numbers? How?
So, I turned to Twitter. Alexander Bogomolny put an explanation on his Facebook page that convinced me my initial 50% hunch was right and helped me see that I wasn’t accounting for the probability that I have to even try a 2nd, 3rd, 4th, etc. time.
Shawn Urban helped me realize why I wanted to be adding, not multiplying, all those probabilities (when adjusted with Alexander’s suggestion).
Earl Samuelson showed another way to confirm that 50% is the correct answer that fit into my most basic understanding of probability: # of favorable outcomes / # of possible outcomes. The total number of possible outcomes is all the ways to choose 50 boxes out of 100 (100 C 50 on a graphing calculator). The total number of successful outcomes are all the outcomes where you pick the 1 box with your name in it from its 1 location (1 C 1) and 49 other wrong boxes out of the remaining 99 (99 C 49). So the probability of picking your name is (1 C 1 * 99 C 49) / (100 C 50) = .5, and the probability of not picking your name is 1 – .5 = .5
I really, really love it when multiple solutions and ways of reasoning through a problem yield the same result. I often learn something about deeper patterns in math when that happens.