Deal or No Deal, Mr Hall! – How We Misunderstand Probability

Recently I was listening to a episode of a great podcast series called Choiceology. It is about how we make choices especially focusing on the factors that lead us astray in those decisions. The specific episode dealt with how framing decisions in terms of potential loss leads us to different decisions than framing them in terms of potential gain.

The episode started with an interview of a contestant from the game show Deal or No Deal. For those not familiar with the show the premise of the game is that a contestant is presented with 26 briefcases containing hidden prize amounts from one penny to one million dollars. Each value is represented only once. After picking one case for themselves the contestant moves through a series of rounds of eliminating cases to reveal their values. Periodically, a mysterious figure known as the Banker makes a cash offer to the contestant to quit the game. The game ends when an offer from the Banker is selected or the contestant eliminates all the other cases and takes the prize in their case.

Probability and Choice

What struck me when listening to the podcast was not the show’s intended theme of framing losses, but another issue that leads us to bad choices. It is our inability to understand probabilities. Luis, the Deal or No Deal player, is one of the few contestants who actually went all the way through the game by turning down every one of the Banker’s offers. The last round had him facing a decision based on two remaining cases in play. Luis either held a case with \$5 or a case with \$750,000. The banker offered \$333,000 to quit.

Take a moment to consider what you would do if faced with this decision?

Here’s the piece that caught my attention. When asked about his decision making process, Luis said that he had a 50/50 chance of having the \$750,000 case since there were only two left. When I heard that statement I knew he was wrong. The probability was not 50/50 that he had the \$750,000 case. In fact, it was still 1 in 26. Confused? The reason I understood the error immediately was because this situation reminded me of a classic puzzle called the The Monty Hall Problem.

The Monty Hall Problem

Premiering in 1963, Let’s Make a Deal was a classic game show that encouraged audience members to dress up in crazy costumes to play silly games. A classic game involved asking a contestant to pick between a \$100 bill or choosing sight unseen what was behind Door #2. That door might conceal a car, a washing machine, or a “zonk” prize like a goat. The show has lived on in many versions with its most famous host being the entertainer Monty Hall.

The Monty Hall Problem is a probability puzzle that continues to confound people to this day. It was originally created in 1975, but became famous after being answered correctly in Marilyn vos Savant’s Parade magazine column in 1990. An army of readers assailed her logic as wrong in harsh and demeaning ways. Her incorrect readers eventually ate crow, or was it goat, and admitted their error after she wrote three columns explaining the logic. The problem plays on our basic misunderstandings of probability.

Here is the puzzle. You are a contest on Let’s Make a Deal. Your game is to pick one of three doors. Behind one is a new car and behind the other two are goats. (For purposes of play assume you want the car.) Monty Hall invites you to pick a door, so let’s say you choose Door #1. Monty, who knows what is behind each door, opens Door #3 to reveal a goat. Then he gives you the opportunity to switch your choice to Door #2. The question: Are your chances of winning the car better, worse, or no different if you switch doors?

Most people when asked this question assume that it will make no difference. After Door #3 is eliminated they assume their chance of winning the car is 50/50. In truth, you are MORE likely to win by switching doors. In fact, Door #2 now has a 66.7% chance of having the car!

The Monty Hall Problem Explained

Let’s start at the beginning. When making the initial selection, the contestant has a 1 in 3 chance of picking the door with the car. That means the contestant also has a 2 in 3 chance of picking the wrong door. After Monty Hall opens one of the goat doors, most players recalculate the odds of winning by shifting the probability from Door #3 between the remaining doors. This is the fundamental error!

Let’s change the format to demonstrate by instead looking at the classic game of three cups and one ball. You are tasked with picking the cup hiding the ball after the host shuffles them around on a table. Assume you choose Cup #1. Now imagine the host pulls out a bucket which he places over the other two cups. He removes the smaller cups under the bucket and tosses them aside. He then asks if you want to keep your original choice or switch to the new improved and oversized Cup #2. It is obvious that you would switch to the bucket to improve your odds to 2/3rds instead of 1/3rd.

That is essentially what happens in the original Monty Hall Problem. By opening Door #3 in our example Monty has effectively placed a bucket over the two doors and combined them into one. The odds from Door #3 only shifted to Door #2, and nothing went to Door #1. Switching improves the odds because the door with the car never changes. The original probabilities after the first choice remained the same. Don’t believe it? This page explains the problem and contains a chart which maps out each possible choice in the game and its outcome. Jim, the author of the post linked to above goes on to explain the reason people mess it up:

The probability that your initial door choice is wrong is 0.66. The following sequence is totally deterministic when you choose the wrong door. Therefore, it happens 66% of the time:

1. You pick the incorrect door by random chance. The prize is behind one of the other two doors.
2. Monty knows the prize location. He opens the only door available to him that does not have the prize.
3. By the process of elimination, the prize must be behind the door that he does not open.

Because this process occurs 66% of the time and because it always ends with the prize behind the door that Monty allows you to switch to, the “Switch To” door must have the prize 66% of the time.

Deal or No Deal Mr. Hall!

Let’s go back to the Deal or No Deal situation. The contestant Luis is facing a type of Monty Hall Problem. His initial chance of selecting the case with \$750,000 was very low, only 1 in 26. Because the prize amounts in the cases do not randomly change the probability from the initial choice remains the same irregardless of how many cases are eliminated. When Luis reached the final choice with only two cases left his assumption of a 50/50 chance was wrong. Since 24 of the 26 cases had values lower than \$750,000 and over half of the cases have values of less than \$1000, his odds of picking a low dollar value case were extremely high. That probability never changed even as case after case was eliminated.

Revealing lower valued cases does not change that probability because we already know that these cases existed. Since Luis only got to pick one case, we know there are 25 other values he did not pick. Luis always had only a 2 in 26 chance of selecting a case with \$750,000 or higher, a 1 in 13 chance. That is immensely different from his assumed 50/50 misunderstanding. The safest option to get the best value in the final round was to accept the sure thing offer from the Banker of \$333,000. If you want to understand the logic of the show better, read this interesting analysis of Deal or No Deal.

Why the Monty Hall Problem Matters

What’s the point of all this analysis you may ask since the odds of becoming a contestant on either game show is very low. The takeaway is that our brains are horrible at understanding probability, yet we live in a world where random chance is rampant. Think about the lottery. CNBC in an article from 2019 stated the following:

The odds of winning the Powerball jackpot are 1 in 292,201,338. That’s a bit better than your odds of winning the Mega Millions jackpot, which stand at 1 in 302,575,350, according to the New York State Lottery. The overall odds of winning any Powerball prize are 1 in 24.87.

The odds of winning even a small prize are so mind boggling that no one in their right mind should play the lottery. Yet every week millions of people play lotteries like Powerball confident that they have the winning numbers. Lotteries only exist because our misunderstanding of probability, combined with the dream of winning big, clouds our judgement.

We face a whole host of probabilistic situations throughout our lives. For example, is it worthwhile to buy extended warranties on appliances? To make an informed decision we have to know the probability that the appliance will fail and the projected cost of an uncovered repair. How about the weather? The accuracy of a five day forecast is much lower than tomorrow’s forecast. Yet we often treat the two forecasts with the same value. Is it a better investment to buy stock in a single company or purchase a market index fund? The truth is that individual stocks are statistically more volatile than a weighted market index, yet most of the business world articles focus on finding the home run stock pick.

Understanding probability is a worthwhile study in our complicated world. For a limited time, Harvard is offering a free course on probability. However, there are many other books and resources available to explain the topic. One of my favorite’s is The Drunkard’s Walk by Leonard Mlodinow.

Finally, if you want to have fun at your next dinner party or socially distanced Zoom get together, challenge people to solve The Monty Hall Problem. It is sure to keep the conversation lively for a long time.