# The intuitive Monty Hall problem

1. [4]
Algernon_Asimov
I would say that the best answer on that StackExchange page is the top-listed one which relies on the host's knowledge of what's behind each door. You know that Monty is going to eliminate the...

I would say that the best answer on that StackExchange page is the top-listed one which relies on the host's knowledge of what's behind each door. You know that Monty is going to eliminate the lousy prize because he wants you to believe that the good prize is still in play: when you pick Door 1, Monty isn't going to reveal the luxury car behind Door 2, he's going to reveal the goat behind Door 3. So you know whatever is behind the remaining door you didn't pick is better than what's behind the door he revealed.

1. [2]
nothis
Yea, I think there's actually some rhetoric smoke bomb in the way the problem is presented that's more confusing than the actual math. You intuitively think the host also guesses, like a proper,...

Yea, I think there's actually some rhetoric smoke bomb in the way the problem is presented that's more confusing than the actual math. You intuitively think the host also guesses, like a proper, fair-playing opponent, in which case he could accidentally open the winning door (or "remove" it from the game, whatever). Here's an explanation what would happen if the host didn't know, suggesting it would put the chances back at 50/50.

It's basically one of those cheap scams, where you overwhelm someone with a common sense scenario but quickly change one element in your favor ("I put \$10 in this envelope, you put \$10 in this envelope, there's now \$20 inside but you can have it for \$15!"). It's more psychology than combinatorics, in the sense that it disguises as a more natural scenario and tricks us into using common decision making techniques in an uncommon situation.

1. Algernon_Asimov
I had to think about that twice to figure out what the catch is!

("I put \$10 in this envelope, you put \$10 in this envelope, there's now \$20 inside but you can have it for \$15!").

I had to think about that twice to figure out what the catch is!

1 vote
2. Kraetos
Yeah, this boxer example helps a little, but it is still a little disconnected from the actual problem to be a "silver bullet." I've seen all sorts of elaborate ways to explain the Monty Hall...

Yeah, this boxer example helps a little, but it is still a little disconnected from the actual problem to be a "silver bullet."

I've seen all sorts of elaborate ways to explain the Monty Hall problem... simulations, 100,000 doors, etc., but in my experience nothing works as well as simply pointing out that Monty knows what's behind the doors.

2. Sahasrahla
An intuitive version I like is to increase the number of doors. Let's say there are 1,000,000 doors and one of them has a prize. You pick a door at random. The host then opens another 999,998...

An intuitive version I like is to increase the number of doors. Let's say there are 1,000,000 doors and one of them has a prize. You pick a door at random. The host then opens another 999,998 doors and leaves #708,174 and your choice closed. Do you stay with your same choice or pick the other door? I think most people would feel like it was intuitive to switch because your original choice was an arbitrary one-in-a-million shot but why did the host leave that door closed?

Some other ways to think of it with this same example:

• You can play this game as many times as you want and every time you play it (whether you picked right on the first try or not) the host will open another 999,998 doors leaving just your choice and another door closed. Does this mean that every time you play the game there's a 50/50 chance the first guess you made was right? How are you so good at one-in-a-million guesses?
• Your friend Bob picks a door. You can either choose to guess the prize is behind his door or one of the 999,999 others. You obviously pick all 999,999 other doors. The host decides to build suspense by opening your doors one at a time (leaving the prize one, if it's there, for last). You know it will take a while because while the prize is almost certainly behind one of your 999,999 doors the host will still have to open the 999,998 doors with nothing behind them. Eventually, after a long wait, there's only one of your doors left to open and the host asks Bob if he wants to switch. Do you think he should?