The Monty Hall Problem

In case you were wondering, The Monty Hall problem is a maths puzzle named after a game show host – first of all, here’s the problem.

A game show host offers a contestant 3 doors – behind one of the doors is a car and behind the other two a goat. The contestant chooses a door; the host then opens one of the other two doors revealing a goat and asks the contestant whether they want to switch their choice to the other closed door. What would you do?

Most people’s immediate response is to stick with their original choice of door on the basis that there is nothing to be gained by switching. However, the fact is that you double your chances of winning if you switch. Now, if you are shouting at your computer, telling me I’m wrong, then you are in good company – plenty of PhDs insisted that there was no benefit to switching when this was first discussed in Parade magazine back in 1990. We’ll get to the proof in a minute (and an Excel simulation) – but first, what has this got to do with Brexit? Well, some of the reasons postulated to explain why people would rather stick with their original choice are:

1) Status Quo bias (a preference for the current state of affairs);
2) The endowment effect (where people value something more that they own, than something they don’t); and
3) The errors of omission vs errors of commission effect (people prefer making errors through inaction than through having taken explicit action).

Brexit voting biases?

So this got us thinking – to what extent are these biases at play in the Brexit debate?
If they are influencing voters, these biases might apply by:

1) voters preferring the status quo of remaining in the EU rather than the ‘unknown’ alternative;
2) voters placing a higher value on our current membership of the EU than on our ‘independence’; and
3) if a voter’s choice turns out to be a bad one, they would rather it were bad as a result of not voting at all (inaction) than of voting to leave (taking explicit action); a vote to remain might also be argued as constituting inaction.

Whilst not so relevant to the game show problem, other biases might also be at play, for example optimism bias, where a voter believes they will be less at risk than other people of suffering a negative event.

We’ll leave you to make up your own mind about that, whilst we get back to the maths:

Solution to the Monty Hall Problem

When you select one of the three doors initially, you have a 1/3 chance of picking the car. That leaves 2/3 chance of the car being behind one of the two remaining doors. These chances don’t alter when the host opens one of the other two doors revealing a goat. This means you now have a choice of sticking with your original choice (1/3 chance of it revealing a car) and switching to the other closed door with a 2/3 chance that it will reveal a car. The reason it is 2/3 is that the door the host has opened will always have a goat behind it so has a 0% chance of containing a car, meaning that the 2/3 chance that was previously allocated to those two doors has to sit with the third, unopened door.

If you are still not convinced, we have built an Excel simulation to illustrate the point which uses a sample of 100,000 examples of the problem, recording the results of choosing to stay or switch. This demonstrates that switching results on average in a 66.7% success rate, compared to 33.3% success rate when sticking. Watch a video that illustrates the problem and explains how the simulation works here:

Or if you would like to examine the simulation yourself, click on the Icon to the left to download it:

Download Monty Hall Excel