Probability and assumptions – The monty hall problem

Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say A, and the host, who knows what’s behind the doors, opens another door, say C, which has a goat. He then says to you, “Do you want to pick door B?”

Is it to your advantage to switch your choice?

Under the standard assumptions, contestants who switch have a 2/3chance of winning the car, while contestants who stick to their initial choice have only a 1/3 chance.

This is the Monty Hall problem. Named after the host of the game show; Let’s make a deal.

It is a math brain teaser that has stumped many great PHD mathematicians such as Paul Erdős , because it is counterintuitive and an example of a veridical paradox.

“A veridical paradox produces a result that appears absurd but is demonstrated to be true nevertheless.”

There is also another way at looking at the problem. The problem is based under the assumption that we have no other information, except what is right in-front of us. It puts an immediacy on an answer that makes projections difficult. What if we look at every episode of the game show in the past. Where are the goats most often situated? What is the distribution of the goats? What is the distribution of the cars?

Next, lets look at the problem like a poker player. The game show host knows where the car is. Looking back at every previous game, what are their tells? Do they do anything differently or slightly that could indicate what door the car is behind?

So why is a math problem from a 1970’s game show relevant?  It’s the last two things I talked about. The historical distribution and the tells.

If we look at the probability of a risk, how can you quantify it? The easiest way is looking at historical distribution, or sets of data. How often has is happened in the past? What where the specifics of the situation? Do those specifics apply to the current environment in question? By knowing this information, you are getting a much more reliable probability than simply guessing, or doing a blind statistical probability puzzle.

Now for the tell. This can be considered a key risk indicator in this scenario. If a certain action happens (A) that correlates to another action happening (B) with consistent frequency, then an assumption can be made that if action (A) occurs, that action (B) is to be expected.

There is an inherent flaw with this theory in that correlation is not causation. Action (A) did not cause action (B), therefore it is possible that they are not related, but rather they are correlated by mere coincidence. However, the more KRI’s (tells) that are used to speculate on the probability of action (B), the higher the reliability of the assumption.

If you know what has happened in the past, you have a better chance of predicting the future.

If you can see the related events, then you can get a feel for when the predicted future will occur.

Some people can feel an ache in their bones when the barometric pressure changes and they know rain is coming. Ask good stock traders how they know what will happen in the market. They look at the trends, the historical data, the cycles. They looks at the signs and they know that what happens in one area will have an effect on another.

The monty hall problem says solve the problem with the information you have.  I say do your research first and check your KRI’s. Sniff the doors for goats and drive away in your car.