In my recent post describing a Monty Hall activity in my AP Statistics class, I shared an amazingly crystal-clear explanation of how one of my new students conceived of the solution:

If your strategy is staying, what’s your chance of winning? You’d have to miraculously pick the money on the first shot, which is a 1/3 chance. But if your strategy is switching, you’d have to pick a goat on the first shot. Then that’s a 2/3 chance of winning.

Then I got a good follow-up question from @SteveWyborney on Twitter:

Returning to my student’s conclusion about the 3-door version of the problem, she said,

The fact that there are TWO goats actually can

helpyou, which is counterintuitive on first glance.

Extending her insight and expanding the problem to any number of doors, including Steve’s proposed 1,000,000 doors, the more goats one adds to the problem statement, the more likely it becomes to win the treasure with a switching doors strategy. This is very counterintuitive, I think.

For Steve’s formulation, only 1 initial guess from the 1,000,000 possible doors would have selected the treasure–the additional goats seem to diminish one’s hopes of ever finding the prize. Each of the other 999,999 initial doors would have chosen a goat. So if 999,998 goat-doors then are opened until all that remains is the original door and one other, the contestant would win by not switching doors iff the prize was initially randomly selected, giving P(win by staying) = 1/1000000. The probability of winning with the switching strategy is the complement, 999999/1000000.

IN RETROSPECT:

My student’s solution statement reminds me on one hand how critically important it is for teachers to *always* listen to and celebrate their students’ clever new insights and questions, many possessing depth beyond what students realize.

The solution reminds me of a several variations on “Everything is obvious in retrospect.” I once read an even better version but can’t track down the exact wording. A crude paraphrasing is

The more profound a discovery or insight, the more obvious it appears after.

I’d love a lead from anyone with the original wording.

REALLY COOL FOOTNOTE:

Adding to the mystique of this problem, I read in the Wikipedia description that even the great problem poser and solver Paul Erdős didn’t believe the solution until he saw a computer simulation result detailing the solution.