The Monty Hall problem comes to the cover of HBR

Nick IngramThinking14 Comments

I’ve just received my latest Harvard Business Review (May 2015) – with a special focus on decision making. But it’s the cover (see above) that I want to blog about today. The three doors, numbered 1 to 3 are clearly a reference to the Monty Hall problem: but none of the articles in the magazine seem to mention it. What is the Monty Hall problem? Well I’m glad you asked!

The problem set-up

The Monty Hall problem is a classic in decision making and statistics. It’s meant to come from the 60s American game show, “Let’s Make a Deal” and is named after its original host, Monty Hall. It goes like this:

  • A contestant is faced with three closed doors. Behind one is a prize. Behind both of the others is nothing. If the contestant chooses the door with the prize behind it, she gets to keep it.
  • The contestant makes her choice, let’s say “I’ll take Door 2”.
  • Instead of opening Door 2, Monty Hall opens another door (either Door 1 or 3 in this example).
  • The door Monty Hall opens will not have anything behind it. (Monty Hall knows which door hides the prize).
  • Say Monty Hall opens Door 3, and reveals that it contains nothing.
  • Monty Hall then offers the contestant the chance to switch her choice.
  • The problem is simply stated – should the contestant switch her original choice? In other words, in this example, should she switch from her choice of Door 2 and choose Door 1 instead (the other unopened door)?

Should the contestant switch her original choice?

Have a think about this before reading any further – see if you can work it out.


 

The solution

So should the contestant switch?

Let’s have a look at one possible case and then see if we can draw conclusions from that. The diagram below illustrates the situation where the contestant chooses Door 2 and then what happens if she switches.

monty hall explanation

The diagram shows the following:

  • The contestant chooses Door 2.
  • At this stage she has a 1 in 3 chance of having made the right choice.
  • The diagram splits into three to examine the three distinct possibilities – the prize is behind Door 1, or Door 2, or Door 3.
  • If the prize is behind Door 1, Monty Hall must open Door 3 (he can’t reveal the prize, and the contestant has already said she wants Door 2). In this case, if the contestant switches, after Monty Hall shows that there is nothing behind Door 3, she wins!
  • Similarly if the prize is behind Door 3, Monty Hall must open Door 1. If the contestant switches (this time to Door 3, her only other option apart from sticking with her original choice of Door 2), she wins!
  • But, if the prize is behind Door 2, Monty Hall can open either Door 1 or Door 3. Regardless of his choice, though, if the contestant then chooses to switch, she will lose.

What the diagram demonstrates is that if the contestant applies the rule “I will always switch my choice” her chances of winning go from 1 in 3 to 2 in 3. Why? Because in two of the three possibilities shown above she will always win if she switches. Whereas, she will only win in one of the three possibilities above if she applies the rule “I will always stay with my first choice”. (I hope you can see that similar diagrams apply equally if she makes initial choices of Door 1 or Door 3.)

In other words, it’s best for the contestant if she always chooses to switch!

The Monty Hall problem is an example of Bayesian Statistics

The Monty Hall problem is a classic example of the application of Bayesian Statistics. This is a branch of statistics invented by an English clergyman a couple of hundred years ago. It deals with how you make decisions as more information is revealed after an initial action is taken.

 

If you enjoyed this post, you can subscribe to my blog so you never miss an update. Click on the “Subscribe” button on the top right of the screen. I post once to twice a week, so your inbox won’t get flooded. Please leave a comment below and tell me whether you got the the answer that it’s always better to switch. Most people (about 80% to 90% apparently) intuitively think the contestant should keep her original choice. You can see why, but Bayesian Statistics tells you that when you get more information you should reevaluate.

14 Comments on “The Monty Hall problem comes to the cover of HBR”

  1. It seems somewhat counter intuitive, but I guess it (once again) shows how important it is to keep learning. Thanks..

  2. Great post! I remember seeing this in a movie (21) but could never understand it’s application. Next time I’m on a game show, I’ll remember this. ;)

  3. Great post! Is it not also about the intentions of the judge? Why whould he open the door if he kew she selected the right door to win the prize? If he wants her to win then she should switch. Can we determine the intention of new information before we decide what to do next?

  4. This problem has always bothered me because opening the first door doesn’t change where the prize is, nor how right or wrong the contestants original guess is (assuming the game is played fairly!).

    If you apply it to business or life, I would not change my mind because the original decision would be made based on the information at hand at the time. If the only thing that is now different is that we know its not behind the first door, it shouldn’t change how I came to the original conclusion. I would only change the door if new information comes to hand that will allow me to make a more informed decision.

    The moral of the story (from my perspective) is, make your decision based on the information that you have at hand. If you get the opportunity, take the time to review your original decision if you have more information at hand.

    1. Nick Ingram

      Hi Michael – great to hear from you! This is right – intuitively the Monty hall Dilemma is a real paradox. The moral you take is a good one – review when more information comes in. The fact that Monty Hall is opening another door is effectively giving the participant new information – but that’s not obvious. I wonder if in business we get new information that we don’t recognise as such?

  5. Great post Nick! I wonder how this would change the decision in the following examples:
    1. The contestant knows that the prize is nominal.. Say $1000
    2. The contestant knows the prize is $1 million.

    I suspect they would be giving away far more $1000 prizes due to removing the pressure from the situation and therefore going against your instinct to stick with your first answer.

    My other question is this.. Given that it’s simply maths, did the show only last for a week or so before people cottoned on?

    1. Nick Ingram

      Yeah, Jarrod, really interesting idea about the pressure. I suspect that the thinking biases that drive us to stay with our first choice are magnified by pressure. So they probably do give away fewer big prizes. I’m thinking of blogging about those biases today – certainly the “endowment effect” is involved, I think. As to the longevity of the program itself, my understanding is that they did other games as well, not just this one. But, interestingly,I’ve seen a study on this. Human beings don’t tend to learn to switch, even over repeated events. Funnily enough, this study also showed that pigeons learnt to switch faster than humans. It seems we over-think things – and come up with the wrong conclusion. The pigeons just mindlessly learnt that switching was better: the humans continued to rationalise.

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