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Alternatively, the way I think of it is as follows: An initial random guess has a 1 out of 3 chance of being correct. Being shown a goat doesn't change this pro

Alternatively, the way I think of it is as follows: An initial random guess has a 1 out of 3 chance of being correct. Being shown a goat doesn't change this probability. Staying with the first choice doesn't change this probability. Only making another choice, and choosing from the smaller pool of two doors increases the odds. Or perhaps better stated in the Wikipedia article: > "By opening his door, Monty is saying to the contestant 'There are two doors you did not choose, and the probability that the prize is behind one of them is 2/3. I'll help you by using my knowledge of where the prize is to open one of those two doors to show you that it does not hide the prize. You can now take advantage of this additional information. Your choice of door A has a chance of 1 in 3 of being the winner. I have not changed that. But by eliminating door C, I have shown you that the probability that door B hides the prize is 2 in 3.'"

dllthomas 7y ago
But being shown a goat can change the probability. If Monty picks randomly (which is not the original problem but is interesting to consider) then there is no benefit to switching.
- rags2riches 7y ago
  Monty picking randomly adds a third outcome. You then have a) Staying wins (Monty opens to a goat) b) Swap wins (Monty opens to a goat) c) First pick lost, no chance to swap (Monty opens to the car) each of equal probability.
- leepowers 7y ago
  Very true! It depends on the particular variation[1] of the game being played. Which is one difficulty of MH - the ambiguity. Probabilities change when the initial conditions are slightly different or when the problem is interpreted differently. I've never assumed Monty picks at random. But it's entirely possible someone else would make that assumption, in turn changing the optimal strategy. [1] https://en.wikipedia.org/wiki/Monty_Hall_problem#Variants https://en.wikipedia.org/wiki/Monty_Hall_problem#Variants

dllthomas 7y ago

But being shown a goat can change the probability. If Monty picks randomly (which is not the original problem but is interesting to consider) then there is no benefit to switching.

rags2riches 7y ago
Monty picking randomly adds a third outcome. You then have a) Staying wins (Monty opens to a goat) b) Swap wins (Monty opens to a goat) c) First pick lost, no chance to swap (Monty opens to the car) each of equal probability.
leepowers 7y ago
Very true! It depends on the particular variation[1] of the game being played. Which is one difficulty of MH - the ambiguity. Probabilities change when the initial conditions are slightly different or when the problem is interpreted differently. I've never assumed Monty picks at random. But it's entirely possible someone else would make that assumption, in turn changing the optimal strategy. [1] https://en.wikipedia.org/wiki/Monty_Hall_problem#Variants https://en.wikipedia.org/wiki/Monty_Hall_problem#Variants