Monty Hall Game

Posted Aug 2, 2004
Last Updated Feb 6, 2014

monty hall gameI was just recently introduced to the Monty Hall Game paradox by my friend Andrew Penry. When he first proposed the game to me I thought it was simply absurd—and my intuitive thinking process would not allow me to accept the statistical conclusions that the game entails.

The game is simple. A dealer puts out three cards, one of which is a winning card. The player may choose a card at random but may not see the card. The dealer then shows the player a losing card that the player did not choose… and the player is given the choice to keep his first choice or switch to the remaining card.

Intuitively, it seems like this is a game of fifty-fifty odds—that you can flip a coin and make a good statistical decision. But that is actually not the case. Empirical statistics prove that you are twice as likely to win when you switch cards than if you keep your original choice.

My mind would not let me believe it. I went so far to write my own JavaScript-driven program to beat it into my brain that this is the truth.

The reason for the mental block is that our brains are psychologically unable to see pure statistics. We assume that because the choice comes to a fifty-fifty decision, it really is fifty-fifty. But Andrew set me straight by explaining something to me.

Assume that the dealer does not show you the one losing card. In that case, the dealer is saying to you, "Keep your card or choose the other two cards.” Even though you know it can’t be both of those cards, you know there is a 66 percent chance that you will win by switching.

It’s crazy, it hurts your head… but over the long run you are always better off by switching your choice.

Go ahead... play the Monty Hall Game and challenge yourself to understand it.

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Mark Wallner

Apr 15, 2009

Hi Shawn,

I'm afraid the logic of your explanation is flawed; if the dealer does not show you a losing card after your first guess, and you switch choices, your chance of winning is not 66% unless you can choose BOTH other cards at once. In that situation, you still know nothing about any of the cards, and each still has a 33 1/3% chance of winning, as far as you know.

In the Monty Hall game, the key is that the dealer knows which is the winning card (an assumption nobody ever bothers to explain when they posit the situation).

I can't remember the exact explanation my dad once gave me, but it has something to do with the dealer's choice telling you something new about the third card, changing the odds on that one to 50%, while the odds on your first choice stay at 33 1/3%.

I think the explanation is that the dealer knows which two cards are losers, and he can't choose your card because he wants the game to go on, so the fact that he didn't choose card #3 MAY be because he knew it was the winner. I can't figure how to assign odds to this situation though.


Shawn Olson

Feb 7, 2014

Well the article does say that if the dealer doesn't show you a loser you still get to choose the other two cards at once. You are correct that if you cannot choose two cards then each card is at 33%.

Even though the dealer does know the winning card, and that certainly factors into the scenario, I think that what is most important about this game is that it shows very clearly that our intuitions are easily tripped up.

In this case, the first card picked had 33% of being correct and new information did not change that original odd. Only after the fact do you know the actual values (by getting the information). The point of statistics is to try and extrapolate what a value is likely to be. Anyone knowing or not-knowing what card is what did not change that initial statistical value--which itself was a construct of the game.

Thanks for the comment! And sorry it took me so long to respond. I somehow didn't see the comments here until a few years after posted :/

Foo Bar

Sep 9, 2008

While you're at it, make a version where you can play with more than three cards. It might be easier to see what's happening playing with ten cards and seeing the dealer eliminate eight goats before asking you to switch.

Shawn Olson

Feb 7, 2014

I may look into something like that at some point. Thanks for the suggestion.

Velexia Ombra

Apr 23, 2008

I think something is wrong with your code... I have tried this on other websites and received 4/6 for staying and 7/3 for switching.... but on this one I am 17/17... I was also 10/10 and 5/5.

Shawn Olson

Feb 7, 2014

That can just happen at times... or it could be the JS engine on the browser you are using. I've never had those results and always end up at 33% or 66% depending on strategy if you go long enough.
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