I still do not believe the "correct" solution to the Monty Hall Problem.
Here is my reasoning:
The player can pick from $1$ of $3$ doors.
The prize can be behind $1$ of $3$ doors.
Monty will open $1$ of $3$ doors.
$3 \times 3 \times 3 = 27$ possible sequences of events.
In $15$ of those possible events, Monty either opens the door the player picked or the door with the prize. Since Monty will not do either of those, these $15$ events are removed from the possibilities.
Of the remaining $12$ possibilities, $6$ times the player wins if he stays with his original pick and $6$ times he wins if he switches. It looks to me like the player has the same chance of winning whether he stays or switches.
Could someone please explain the flaw in my reasoning.
Best Answer
The basic flaw in the reasoning is that it assumes that each of the 12 possible outcomes are equally likely -- because without such an assumption "same number of outcomes" does not translate into "equal probability".
In many simple probability situations we can get a plausible "all outcomes are equally likely" assumption from symmetry arguments. But that won't hold here because the group of outcomes where the contestant chose the prize door are fundamentally different from the group of outcomes where the contestant chose a non-prize door; the host have different amount of options in the two cases.
As a simpler illustration that it doesn't always work to assume that all the possibilities are equally likely, consider this different game:
Here there are a priori 12 outcomes: {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}. But some of them are impossible; if we exclude them there are only 7 outcomes: {H1, H2, H3, H4, H5, H6, T6}. In 6 of the cases the coin flip was head; only one cases was the tails.
Can we then conclude the odds of flipping heads is 6 to 1?
Of course not. The coin is still fair, and what happens after it is flipped can't influence the probability of coming up tails. So this shows that comparing numbers of outcomes doesn't always give true probabilities, which ruins your Monty Hall analysis.