Yup, I made a mistake. In my calculations of John Reemtsen's odds of
winning three times out of five spins of a wheel to get way more steak
than he could possibly need, I forgot a crucial detail. Huge thanks to
David Weinberg, chair of OSU's Department of Astronomy, for pointing
me in the right direction.
I was correct in figuring that there were 10 different arrangements of
winning three times out of five. For example, he could've won on his
1st, 2nd, and 3rd attempts, or on his 2nd, 4th, and 5th attempts, and
so on. And the process for figuring out John's odds are still the
same: count up the total number of ways he could win, and divide that
by the total number of all possible combinations.
So far, so good. But here's where I miscounted: there were 32
positions on the wheel, which means there isn't just one way to lose,
but 31! Let's say John won on his first three attempts. For the next
two spins, there are 31*31=961 ways to *not* get the steaks.
This means that there are many more than 10 arrangements of wins for
John to get his juicy steaks; there are 10*31*31, or 9,610 possible
winning combinations. This increases his odds of winning from one in a
million to one in 3,491.
John is still very, very lucky, but maybe he doesn't owe me a free
steak any more.