The Monty Hall Problem: It’s a Piece of Cake

In chapter 1 of Harvard psychology professor Steven Pinker’s new book, Rationality: What It Is, Why It Seems Scarce, Why It Matters, Pinker discusses the so-called “Monty Hall problem”, which concerns the tricky probabilities involved in a game-show that is similar to the old TV game show “Let’s Make a Deal”.  In the game, there are three numbered doors, and a prize exists behind only one of the doors. The player chooses one of three numbered doors. Game host Monty Hall – who knows behind which of the three doors there is a prize – then eliminates one of the two remaining doors. However, the rules of the game do not allow Monty to eliminate a door having a prize behind it. Monty then asks the player whether she would like to trade her door for the one remaining door.

The Monty Hall problem asks: To maximize her chance of winning the prize, should the player trade her door for the remaining door?  Interestingly, according to Pinker, most people answer that there is no point in switching doors because – there being only two doors left – the prize is just as likely to be behind one door as it is to be behind the other remaining door (i.e., there is a 50/50 chance of choosing correctly). However, the mathematically correct answer to the question is that the player should trade her door for the other remaining door, because there is a 2/3rds chance that the prize is behind the door Monty did not eliminate, and their remains only a 1/3rd chance that the prize is behind the door initially chosen by the player.

As many people see it – including many reportedly testy PhDs of mathematics – if there are only two doors, and a prize may be behind either one of them, the prize is no more likely to be behind one than behind the other. For most, it is counter-intuitive that the chance of the prize being behind one door is double the chance of it being behind the other. However, I offer the following – an analogy to the game played in the Monty Hall problem – in the hope that it might assist you (as it did me) to find it easier to see that – at least in the Monty Hall scenario – switching one’s door is always advantageous. Specifically, I bring you the game show: “Piece of Cake”.

Rules of the Game

The host, Monty, presents you with big round chocolate cake. He tells you that the cake’s baker dropped his house key into the cake batter, such that the key is to be found somewhere in the cake. The cake is decorated with lines of pink icing that demark 12 equal wedge-shaped areas of cake. The key is hidden within one of the areas. Monty knows which area holds the key, but you do not.

Monty lets you pick any one of the 12 areas of the cake. Once you make a selection, Monty will cut from the rest of the cake the wedge-shaped 1/12th of the cake that you selected. He’ll put that on a plate in front of you.  Let us call your little slice of the cake “your portion” and let us call the rest of the cake “Monty’s portion”.  To be clear, your portion is 1/12th of the cake, and Monty’s portion is 11/12ths of the cake.

Now, at this point, it should be intuitively obvious that the key is less likely to be in your portion than in Monty’s portion.  Consider: if you were given a choice between (a) searching most of the cake for the key, and (b) searching very little of the cake for the key, would you choose to search most of the cake, or just a little bit of it? Your probable answer is: (a) most. Why? Because if the baker’s key could have landed anywhere in the cake, then – all else being equal – the chance that you’ll find it in a large portion the cake is higher than the chance you’ll find it in a relatively small portion of the cake.

The rules require Monty to cut one more 1/12th wedge from Monty’s portion after handing you your portion. However, the rules say that if the key is in Monty’s portion, Monty must cut from the remaining cake only the 1/12th wedge that contains the key.

Monty cuts his wedge. The throws out the rest of the cake.

Then Monty asks whether you would like to switch your slice for his before he reveals which of the two pieces contains the key.  If the slice you ultimately choose holds the key, you will win the baker’s house.

 

Common Sense that Corresponds to the Correct Result?

Now ask yourself: Did throwing out 10/12ths of the cake change the fact that the chance of the key being in Monty’s portion is 11/12?  Is it really hard to believe that the chance of the key being in Monty’s portion remains 11/12, whereas the chance of it being in your portion remains only 1/12?  If Monty cannot throw out a key, and the key was 11 times more likely to be in Monty’s portion than in your portion, isn’t it obvious that you should swap slices with Monty?

Isn’t it now equally obvious that, to win the prize, you should swap doors with Monty in the “Let’s Make a Deal” game?  I hope so and, if you found my key-in-a-cake example to make it obvious, please share it with others who may benefit from it.