Understanding Car, Goat, Pigeon

image of door, car and goat

(reading time: 3 mins)

Understanding something that runs contrary to ‘common sense’ is often about finding the right form of words, in this case The Monty Hall problem.

Here’s a quick description of the problem in case you’re not familiar with it:

You’re the contestant on a TV game-show, trying to win a car. You are given three doors to choose from. The car has been randomly placed behind one door, behind the other two are goats. The host knows where the car is, so once you’ve made your choice, the host opens one of the two other doors to reveal a goat – a ‘wrong’ answer. That door is then discarded. Now you are given a choice; stick with your original choice or pick the other remaining door. Surely it doesn’t make any difference if you change because with two doors your odds are 50/50 whatever you do?

In fact your odds of winning the car are 2 in 3 if you always switch when given the choice. Marilyn Vos Savant who described the problem in Parade Magazine in 1990 and the best strategy, always switch, got a hostile reaction from thousands of readers insisting she was wrong. Some critics had degrees in related disciplines and really should have examined her reasoning properly, instead of digging in their heels and dismissing her arguments.

The problem demonstrates both difficulty of the mind understanding something that goes against ‘common sense’, but also exploits a behavioural tendency in the contestant. Even with training, in tests most people are more likely to stick with their original choice. Wrongly believing their odds have only increased to 50:50 they make the understandable decision to stick with the first choice no matter what, probably to avoid the feeling of additional dis-appointment on losing if they had actually chosen the car first time.

There are plenty of explanations out there as to why the odds improve if you switch, most written in the language of mathematics and probability, which may be too obscure to overcome ‘common sense’. My own way of describing why always switch works, is to realise this is a two-stage process. If you stick with your original choice you are sticking with the 1 in 3 chance you had in the first round. You are effectively choosing not to participate in the second round. However if you do switch you are then participating in a second round where the host, by taking away a wrong answer, is increasing your odds of winning. He is actively directing you towards the right answer. The second round is not the same as a single round game where you only ever had two random doors to choose from, because the host’s intervention in eliminating one wrong answer increases your odds beyond the 50/50.

The Monty Hall problem is however very different from another common misconception about probability – the strategy of picking lottery numbers by favouring those which have occurred least on previous draws. The odds of any one lottery number appearing are the same every time and don’t average out, because each lottery draw is an isolated event. Unlike the game show, lottery draws don’t have a first round then second round, with a favourable intervention in between.

So what’s interesting about those hostile reactions is how easily emotional attachment and reputation can get in the way of something humans often assume makes us the superior species – reason. I’ll bet there was a dose of sexism in some of the criticism Vos Savant recieved. They initially did not understand her reasoning because it ran contrary to the ‘common sense’ view. And some critics did not properly consider her challenge to the ‘common sense’ view because it came from a woman writing in a popular magazine, so were quick to write her off and lump her in with ‘averaging out’ lottery players.

More intriguing still, when Pigeons – the ‘flying rats’ of our cities – are given the same problem, they are quicker to recognize this pattern1, and adopt the always switch strategy than humans. Which means Pigeons can grasp in days what some academics have taken years to understand!


1. Pigeon performance on the Monty Hall problem https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3086893/