There is a famous problem in probability called the Monty Hall problem. The name Monty Hall belonged to the host of a game show, where contestants were presented with three doors – one of which had a car behind it, and the other two had goats. Obviously most people would prefer the car. Contestants were asked to choose a door (which remained closed), and Monty Hall would then open one of the other doors to reveal a goat. After this the contestants were asked if they wanted to swap to the other closed door from the one they had previously chosen. It turns out that the contestant should always swap – it actually doubles the chance of winning the car from 33% to 67%. This is not obvious and if you want a full explanation then look at the following Youtube video.
What is surprising about this seemingly simple problem is that Professors of Statistics, statistics PhDs and a host of other experts all got it wrong, and in fact stubbornly refused to believe the actual outcome until the game was simulated on computer. All the more annoying to the academic elite was the fact that the correct analysis was presented in a popular US magazine called Parade (no, not the Journal of Statistics) by Marilyn vos Savant – a frighteningly smart woman with a record breaking IQ. A senior figure in the US Army commented that if vos Savant was correct and all those PhDs were wrong, then the country was in pretty bad shape. Well it turns out she was right and …
There are a number of seemingly simple problems of this nature (the Twins problem for example), and they all illustrate very well how difficult it is for us to think in probabilistic terms – something that Daniel Kahneman writes about extensively in ‘Thinking Fast and Slow’.
This has implications for the rapid rise of probabilistic methods used in data mining and statistics where everything is never as it seems. It’s very easy to press the ‘go’ button using analytics tools, and rather more difficult to determine whether the analysis actually makes sense. And as for drawing conclusions from the new generation of eye-candy rich visualization tools – well, read this if you are interested.