How to get some money from Monty

How to get some money from Monty

Hey everyone, I’m Hunter and I decided to dive deep into the statistics and reasoning of the Monty Hall Problem . The basis of the Monty Hall problem comes off of a famous game show. The famous show “Let’s Make a Deal” was one of the most popular shows on weekday mornings. The show’s famous host, Monty Hall, would give away a car on the show.

A Brand New Car From “Let’s Make A Deal” Source: Bestride.com

He did it in a simple way: Show a contestant 3 doors. One door had a car behind it, and the other two had goats behind them. He would ask a contestant to choose one door, and then Monty would open one of the doors the contestant didn’t pick, revealing a goat behind the door. He would then pose the question “Would you like to switch doors?” meaning, would the contestant like to stick with their original choice or switch to the third unopened door.

Monty would show them three doors, pictures above Source: Devlin’s Angle

Natural instinct is to stick with your gut and keep the door you originally picked. This is the origin of the Monty Hall Problem. It wasn't until Bayes theorem was applied that the truth of the probability came out. Bayes was way ahead of his time with the creation of the formula to predict the outcome of an event happening based on prior conditions. This is perfect for the Monty Hall problem. The theorem, when applied to The Monty Hall problem, shows that:

• The original chance of getting the car behind the door you choose is ⅓.
• After Monty reveals the goat behind a door, then using the Complement Rule from Statistics, the door revealed is factored out of the equation and all probabilities must sum to 1. Therefore 1-⅓ , is the remaining probability that the other door has the car, or ⅔.
• The famous movie “21” is based on more statistics in card counting and blackjack, but there is a key scene that depicts this perfectly where the switching of the doors gets your a higher probability.

A model of the three doors Source: Statistical Engineering

I used a google survey form to get a sample from my school. I asked them to play the simulation game and then answer if they switched or didn't switch and then if they won or lost. After running the test on 50 upper school students at my school, I got an even split that 25 switched and 25 didn't switch. The results were astonishing high for switching. Out of 25 students 21 won while switching but only 17 won while not switching. This data accurately represents the math behind the problem. If you switch you have a higher chance of winning. In the test run, a 20 percent higher chance.

The common mistake people usually make is thinking: “Oh, once the door is revealed, wouldn't the probability of you having the car be a 50/50 shot now?” This leads to the paradox that the Monty Hall Problem creates. The key is that Monty is improving the chances of the door that you didn't pick being a car by revealing a goat behind the other door. Let me further explain, imagine you have 100 doors, one with a car behind it and 99 with goats behind them and you pick one, hopefully finding the car. You have a 1/100 chance of getting the car. Then, Monty reveals the next  98 doors with goats and leaves 2 left, your door and one other. Would you still consider it a 50-50 chance? You already know that those other 98 doors were thrown out for a reason, from this you can decide to choose the door that is left out. The more information you know the better your chances are.

This same idea puzzled some of the greatest minds in history including one of the smartest women in the world, Marilyn vos Savant. She published the theory on switching having a better chance of winning and was bashed and ridiculed by thousands of people telling her she was wrong. She improved her theory with more work explaining her reasoning with a test of 1,000 schools and sure enough, the data showed her theory to be correct. Vos Savant also was asked the question of the similar “Two-Boys Problem”, which has similar aspects of the Monty Hall Problem. Vos Savant was posed with this very question:

“A shopkeeper says she has two new baby beagles to show you, but she doesn't know whether they're male, female, or a pair. You tell her that you want only a male, and she telephones the fellow who's giving them a bath. "Is at least one a male?" she asks him. "Yes!," she informs you with a smile. What is the probability that the other one is a male?”

Everyone assumes that the events are independent. As they say in statistics meaning the 2 puppies being different genders do not have an effect on each other, or in other words, one being male doesn't make the other a certain gender. This makes the reader of the problem think that the dog has a 50-50 chance of being a boy, just like in the Monty Hall Problem. Once again, Vos Savant proved it wrong. The way to look at is that if there are four possible combinations of puppies: Girl,Girl, Boy,Girl, Boy,Boy and Girl, Boy, then once the man giving them the bath tells you that one is a boy, then the Girl Girl senario is out and out of the 3 scenarios left, only one has the ability to be Boy, Boy therefore the probability is ⅓. The catch came in the question. The reader assumes that the puppy being bathed at the time of the call is a boy then that must mean the second puppy has a 50-50 chance. This table gives the same statistics as the puppy problem, but it instead uses a similar problem of the older and younger sibling problem.

This is the breakdown of the “Two Boys Problem” Source: Wikipedia

This basic theory of probability has puzzled the greatest minds in history. Most of the time it is the human brain overthinking reason and replacing it with emotion. The Monty Hall problem is that specifically. No one wants to switch from their door because of it being a psychological thing. Switching makes your chances better even though your brain tells you that there is a 50-50 chance.

This bar graph shows the results of various tests and seeing the physiological results Source : The Psychology of the Monty Hall Problem: Discovering Psychological Mechanisms for Solving a Tenacious Brain Teaser

I hope this helps untangle the complex web of the Monty Hall Problem. At first, it seems to be super complicated and complex, but when it is boiled down to the statistics, it can be seen that switching will get you some money from Monty or at least on a game show.