The Monty Hall problem comes to the cover of HBR

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.

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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.

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.