Question
What is the simplest, most convincing explanation of the solution to the Monty Hall problem?
Answer
Personally I find the second approach mentioned by Henry Robinson works the best to convince me that the right answer is to switch (the large numbers case).
It still isn't conceptually satisfying to me in causal terms though. For that, the best approach for me has been to think in information-theoretic terms. Probabilities never change unless information flows from one decision-making agent to another.
In this case, by opening a door he knows is not the prize, the game-master is revealing some information. Information is flowing, so probabilities will change and decisions may need to switch.
If, instead of opening a goat door, he'd merely added some irrelevant piece of information to the picture ("I'll give you a hint... my cat's name is Whiskers. Do you want to change your selection now?"), the answer would be different.
Or to bring it closer home, if he'd said "I am going to use this roulette wheel and pick a random number, and open that door, and you can decide whether to switch" then he is adding no extra information. BUT he might have ended up opening the actual prize door in that case (in which case, of course you should switch with 100% probability of winning the prize). The only way he can avoid revealing the prize door is to reveal actual information.
It still isn't conceptually satisfying to me in causal terms though. For that, the best approach for me has been to think in information-theoretic terms. Probabilities never change unless information flows from one decision-making agent to another.
In this case, by opening a door he knows is not the prize, the game-master is revealing some information. Information is flowing, so probabilities will change and decisions may need to switch.
If, instead of opening a goat door, he'd merely added some irrelevant piece of information to the picture ("I'll give you a hint... my cat's name is Whiskers. Do you want to change your selection now?"), the answer would be different.
Or to bring it closer home, if he'd said "I am going to use this roulette wheel and pick a random number, and open that door, and you can decide whether to switch" then he is adding no extra information. BUT he might have ended up opening the actual prize door in that case (in which case, of course you should switch with 100% probability of winning the prize). The only way he can avoid revealing the prize door is to reveal actual information.