Monty Hall hosted the game show “Let’s Make a Deal” for many years. One of the features of the show was to allow contestants to choose one of Door Number 1, 2 or 3 to win a big final prize. It might be a new car or a dream vacation, but it was much more valuable than the junk behind the other doors.

There grew a “paradox” around this selection process that went something like this:

**Monty tells you that there is a brand new car behind Door Number 1, 2 or 3. Behind the other doors are goats. He asks you to pick one of the doors. You pick Door Number 1. To the horror of the sponsors, Monty decides to give you a hint. He knows the correct door, but he will not tell you that. Instead, he tells you that there is a goat behind Door Number 3. He gives you a chance to change your pick. Should you do it? Does it matter?**

Most say that it doesn’t matter whether you change or not. They think that their odds of being correct have just gone up to 50%, but changing to Door Number 2 won’t improve those odds. Even Nobel laureates and statisticians tended to agree with this intuitive answer. They were wrong. You should ALWAYS change to Door Number 2 to increase your chance of success to 2/3!

Why?

This “problem” is often called a “paradox,” but it isn’t. The answer can be clearly demonstrated by computer simulations, physical experimentation and complex Bayesian probability. It is hard to “see” the answer without resorting to some form of demonstration.

Why is it so hard to “see”?

There are really three reasons:

- We are loathe to think through the consequences of our first choice being wrong,
- We are troubled by having to deal with the impact of new information, especially how it impacts the decision path we initially rejected and
- It is hard for us to think about all of our choices resulting in probable outcomes and that our “best” solution is the one that “improves” our probability for “success” even though it does not make the outcome certain.

Because of these issues, we naturally gravitate to simple evaluations and fail to look deep enough into the possibilities to get to the right answer.

The right answer does NOT require a PhD in statistics. In fact, the most educated are often the most recalcitrant in their belief that the choice of Door Number 1 or 2 is of equal probability. Some have gone to great lengths to try to support their intuitive but incorrect conclusions. The solution is pretty simple, but the logic must be followed with care.

**Event 1:**

We pick Door Number 1. Our chance of being correct is 1 in 3 (or 1/3). Our chance of being wrong is 2 in 3 (or 2/3).

**Event 2:**

Monty looks at what is behind Doors Number 2 and 3 and tells us that behind one of them (in this case Door Number 3) is a goat. He gives us a chance to change our choice – obviously to Door Number 2.

Our logical choice will always be to switch to Door Number 2. That will double our chance of being correct from 1/3 to 2/3.

Event 2 does not change our probability of having chosen correctly the first time. It is still 1/3. What we do know, however, if our first choice was wrong and the car was behind Door Number 2 or 3, we now are CERTAIN that it is behind Door Number 2. This is known as assessing a conditional probability – that is figuring the chance of success after some other related event has occurred. The key to our understanding is assessing this second path, that is, the path where our first choice was wrong. It is this evaluation that give us the clue for what to do.

If we were wrong (and we will be wrong 2/3 of the time), then this new knowledge tells us with certainty that the car is behind Door Number 2. If we now switch to Door Number 2 our chances for picking correctly jump to 2/3.

Sometimes if we look at the options graphically it makes more sense:

What we see is that 2/3 of the time we will be going down the path we thought was wrong – that is the car was behind 2 or 3 and not 1. But, when Monty tells us that the car is NOT behind Door Number 3, we suddenly realize that IF we were wrong in our first guess, then the car MUST be behind Door Number 2. Since our first choice would be wrong 2/3 of the time, we can improve our chances of getting the car to 2 out of 3 times by changing our choice to Door Number 2. That is better than the 1 out of 3 we had at the outset. It is even better than some naïve guesses of 50/50 at Event 2.

If you don’t believe these results you are in good company. To prove it to yourself, simulate this with a friend using three playing cards. Try to pick one of the cards (maybe the Ace of Spades?), from among the other cards placed on the table face down. Have your friend look at the other two cards and tell you one of them that is NOT the card you are looking for (i.e. is NOT the Ace of Spades). In about 30 tries you will see that the analysis approximately matches your experience.

The moral of the story is that we often fail to evaluate new data correctly, especially when it appears to be contrary to or disconnected from our original hypotheses.

This is just another difficulty in staying on the path of Evidence-Based Decision-Making.

*Stites & Associates, LLC, is a group of technical professionals who work with clients to evaluate, improve and deploy new technologies by applying Evidence-Based Decision-Making. The Founder, Ron Stites holds a BS in Chemistry and an MBA in Finance and Accounting. He teaches Business and Marketing Research at Colorado Christian University. For more information see: *www.tek-dev.net*.*

## Comments