Stick or Switch Solution

The sensible decision is always to switch because doing so increases your probability of winning from 1/3 to 2/3 compared with 'sticking'.

There are many explanations of this (see but I think the following is the easiest to understand.

It is clear that when you first choose, the probability of choosing the winning door is 1/3.  If you stick to your first choice the probability of winning stays at 1/3;
nothing that Monty Hall says changes this probability.  Since there is a 1/3 probability that your first choice is the prize winning. it follows that the probability that your first choice is NOT the prize winning door is 2/3.  But if your first choice is NOT the prize winning door then you are guaranteed to win by switching doors, To see this, suppose you chose door 1 and that the prize is behind door 2. Then Monty Hall would have to reveal door 3, which has no prize. And if the prize is behind door 3 then Monty Hall would have to reveal door 2, which has no prize. So, since you ALWAYS win by switching in the case where your first choice was not the prize winning doore, there is a 2/3 probability of winning by switching.

It may be easier to appreciate the solution by considering the same problem with 10 doors instead of just three. In this case there are 9 doors without the prize behind them and one door with the prize. The contestant picks a door. Monty Hall then has to reveal 8 of the other doors without the prize leaving just the contestant's door and one other door. Unless the contestant chose the one door with the prize (a 1/10 probability) the only door left unopen by Monty must be the one with the prize behind it.  So there is 9/10 probability that the other door will contain the prize, as 9 out of 10 times the player first picked a door without the prize. The contestant's probability of winning is therefore 9/10 if he swtiches compared to 1/10 if he does not.

The 'trick' about the three-door game, is that it creates a misleading impression;  the contestant is always presented with 1/3 proportions. There is a 1/3 chance of winning, the host reveals 1/3 of the mystery, and the player is allowed to switch to the other 1/3 option. All options seem equal — yet they are not.  Another reason why people overestimate the probability of winning if they stick is that when Monty Hall reveals that a door that they did not choose has no prize it tends to confirm in their mind that their first choice was 'correct'. This is an example of what is commonly referred to as confirmation bias. It is quite irrational of course because Monty Hall must always reveal a door without a prize, irrespective of your choice.

A full probabilstic solution using a Bayesian net that you can run without knowing anything about Bayesian probability can be found by running the model called "Monty Hall Dilemna", which is in the Examples/Basic folder of the AgenaRisk tool.

Norman Fenton

Return to Main Page Making Sense of Probability: Fallacies, Myths and Puzzles