What’s Behind Door Number Three?

by Marc Rounsaville on December 13, 2011

Sometimes we aren’t nearly as smart as we would like to think. Our brains can be full of tricks, just when we think we know how something works or we get a concept “down” we get shown maybe we haven’t mastered it. This is very true in the risk decisions and probabilities.

There used to be a game show on television Let’s Make a Deal,” where contestants were offered choices and then frequently offered a chance to change. The participants get to choose the box or the curtain, at the end of the show someone was selected to play in the final “big game”. Choose the right door out of three and you win a new car.  Not a very complicated premise and not much to learn about decisions or probabilities there is a 1 in 3 or a 33% chance to win a car at the end.  Right? Wrong.

There are a number of subtle but powerful activities going on, working in the background if you will, when facing a decision – simplifying a complex world, settling the tension of not knowing, and seeking security to name a few.  We are going to focus on two things – decision loyalty and conditional probabilities. We will get to those in minute but first let’s go to the game.

In the final game there are three numbered doors. Behind one door is a prize and behind the other two nothing. You can pick any of the doors that strike your fancy. After you pick your door one of the other doors will be opened to show there is no prize behind it. At this point you get to make another decision, keep your original choice or switch. What to do? Most people, the vast majority of people, stick with their original choice. In fact, in one controlled study 100% stayed with their original choice. In another study 87% refused to switch. Why? The odds are now 50:50, right? It is just like flipping a coin, right? In the both controlled studies switching was overwhelmingly the right decision. Now you’re thinking that cannot be. There are two doors, one prize. Odds are even between the 2 choices. Well they are not. In fact switching choices is two times more likely to win the prize than staying with the initial choice. This can be explained by the principle of  conditional probability.

Before we get to the probabilities let’s look at decision loyalty or why people wouldn’t switch. Numerous studies have shown that people are loyal to their choices. Even when there is strong evidence they should switch. Marketing gurus know this as brand loyalty. This is one of the reasons market share is so important to manufacturers. In the game show case there was no obvious reason (at least to those unfamiliar with the nuances of probabilities) to switch. Think about some of your purchasing decisions, the brand of car you drive, television you watch or soda you drink, likely you have been buying or using the same brand for a long time. Shift this context to emergency management or a fire response, go direct or indirect. How hard is it to abandon that initial decision? Same for a strategy at work, PC’s or Macs, direct sales or distributors. It really doesn’t matter the what just know people are loyal to their decisions even though it can take them off the proverbial cliff. The decision is ours, we created it, we own it and we find it hard to abandon.

Now let’s look at the second principle — conditional probability. This particular problem is sometimes referred to as the Monty Hall Problem. There are three doors and one prize. The contestant gets to choose a door. So there is a 33% chance she has selected the door with a prize and a 67% chance the prize is behind the other two doors. The host opens a door without a prize leaving two doors and one prize. Your chances of winning are now 50%, right? Wrong. The chances are still only 33%, nothing has altered the odds. There is still a 67% chance the other door hides the prize. Opening a door didn’t change the probability. Since the two doors together had a 67% chance the one remaining door holds the 67% probability.

You can test this principle quite easily with note cards and a friend. The “host” randomly places a coin under a note card; the “contestant” chooses a card. The host then reveals one location without the coin. At this point the contestant switches their choice and the host shows where the “prize” is. Do this about ten times keeping track of how many times the contestant wins. Run the contest again only this time the contestant doesn’t switch, once again keep score. It is important for both the host and contestant to play straight up and not try to “game” the game. There are more complicated ways to run the trails which would ensure a more random distribution of prizes, though the simple way presented here will yield the expected results. The contestant that switches will win more frequently than the contestant that does not. One caution, ten trails is a pretty small sample and may not be statistically valid.

This is hard to accept, it is counter intuitive and frankly you will be able to find academics and others that will disagree.  The world is complex and complicated and we tend to simplify things so to better understand. Probabilities can quickly become difficult; often they don’t match with our intuition. We sometimes substitute feelings for objectivity. These are impacts to our decision making.

Where has decision loyalty shown up with unexpected results for you?

Has there been a time when the expected outcomes didn’t match with the actual outcomes?

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