Here's a question that occurred to me while I was walking home from the subway recently: What's the average winning roll in a five-man group, assuming everyone rolls? That is, if you randomly chose five numbers between one and 100, what is the expected value of the highest one?
I know a bit of statistics, but I really don't know how to begin getting at that one. However, I do know how to write a script that will calculate the answer. (Yes, these numbers are only pseudorandom, but I did some limited testing with real random numbers (from random.org) and the results were the same. Besides, I'm pretty sure Ruby's Mersenne twister pseudorandom generator is good enough for testing distributions like this.) The average winning roll out of a group of five people is
82.8 83.8 83.3 (tested over many, many repetitions). Now can any mathemagicians tell me why?
The graph above, in case it isn't clear, is average winning roll on the y-axis vs number of people rolling on the x-axis, tested over 100,000 trials for each group (the relatively small sample size is why the first point is not right at 50, and probably why the curve is a little wobbly).
Edit: I made a silly mistake and did my rolls from 0-99 instead of from 1-100. Crschmidt caught the error and provided the answer to my question: "splitting the available pool into 5 evenly distributed rolls would give you that result. (5 splits, 6 sections, 100 -(100/6) == 83.3." Thanks!
Edit 2: Another good explanation by Jeffreyssai: "The average winning roll for n people is (1 - 1/n) * 100 (plus or
minus a little chunkiness because you can only roll integers). It's
a royal pain explaining why with no way of typing an integral sign,
so I'll let someone else do that."