Everything Everywhere Daily: History, Science, Geography & More - The Monty Hall Problem

Episode Date: February 9, 2021

For over 30 years, Monty Hall was the host of the game show Let’s Make A Deal. In the show, they played a very simple game where you would choose one of three doors. This simple game has led to one of the most controversial and public kerfuffles amongst mathematicians, which caused many distinguished mathematicians to have egg on their face. Learn more about the Monty Hall Problem on this episode of Everything Everywhere Daily. Learn more about your ad choices. Visit megaphone.fm/adchoices

For over 30 years, Monty Hall was the host of a game show called Let's Make a Deal. In the show, they played a very simple game where you would choose one of three doors. This game has led to one of the most controversial and public kerfuffles amongst mathematicians, which caused many distinguished mathematicians to have egg on their face. Learn more about the Monty Hall problem on this episode of Everything Everywhere Daily. What if your perceptions about the past were wrong? ThruLine is a podcast that takes you back in time, to uncover the parts of the story that may have gone unnoticed.

It effectively turned day into night. And how it shaped the world now. Time travel with us every week on the ThruLine podcast from NPR. This episode is sponsored by CuriosityStream. If you have even a passing interest in mathematics, even if it was never your best subject in school, then CuriosityStream has many shows to help you understand mathematics without the use of complex equations.

Shows like Mathematics, Nature's Mathematics, and Math numbers are just a few of the many shows which can help you understand mathematics at an armchair level. Prices start as low as $2.99 per month or $19.99 per year. One of the cheapest streaming services available online. If you love to learn, and let's face it, if you're listening to this show, then you clearly do. Then start your subscription by visiting everything-everywhere.com slash curiosity stream or by clicking on the link in the show notes. I often have ideas for episodes about mathematics that I have to reject

because it's very difficult to explain most concepts without visual aids. However, there's one problem that's incredibly simple and can be explained over audio. The problem is this. On let's make a deal, they had a game where a contestant was shown three doors. Behind one door, there was a new car. Behind the other doors, you had a booby prize, like a goat or a chicken. Monty would ask the contestant to pick one of the three doors.

So far, this is pretty easy. You pick a door, and you have a one in three chance of winning the car. However, after you selected your door, Monty would then open one of the two remaining doors, which had one of the booby prizes. After doing so, he would then give the contestant the opportunity to switch doors if they so choose. The question is this. Should the contestant switch doors?

Do your odds improve by switching, or do you? does it even matter? This question was dubbed the Monty Hall problem in a 1975 letter to the journal American Statistician by Stephen Slevin. However, an equivalent problem called the Three Prisoner Problem was published in a 1959 edition of Scientific American. The problem came to the attention of the public and caused a media furor when it was published in a column by Marilyn Voss Savant in the September 9th, 1990 edition of Parade

Magazine. Marilyn Voss Savant was once listed in the Guinness Book of World's Records as having the world's highest IQ. Her answer was that you should always switch doors as it would double your odds of getting the prize. Her answer to the question resulted in a flood of thousands of letters, many of whom were from PhDs and mathematics and were professors at major universities. Here's a sample of some of the letters which came in. Quote, since you seem to enjoy coming straight to the point, I'll do the same. You blew it. Let me explain. If one door is shown to be a loser, that information changes the probability of either remaining choice, neither of which has any reason to be more likely to one half. As a professional mathematician, I'm very concerned with the general public's lack of mathematical skills. Please help by confessing your air and in the future being more careful. Robert Sacks, Ph.D. George Mason University. Another letter was even more adamant. quote, you blew it, and you blew it big.

Since you seem to have difficulty grasping the basic principles at work here, I'll explain. After the host reveals a goat, you now have a one and two chance of being correct. Whether you change your selection or not, the odds are the same. There's enough mathematical illiteracy in this country and we don't need the world's highest IQ propagating more. Shame. Scott Smith, PhD, University of Florida. There were many, many similar letters. However, there were also just as many letters that were sent in supporting her answer. Here's one of them.

Quote, you are indeed correct. My colleagues at work had a ball with this problem, and I dare say that most of them, including me at first, thought you were wrong. Seth Klasson, Ph.D., Massachusetts Institute of Technology. So, who is correct? What is the right answer? The correct answer is the one given by Marilyn Voss Savant. Your odds would be better if you switch. There are several ways to look at the problem.

The key is understanding that your odds didn't change when Monty opened one of the doors. When you made your initial choice, you had a one-and-three chance of being right. That means the other two doors had a two-and-three chance collectively of having the prize. However, there is a 100% chance that one of the two doors you didn't pick has a booby prize. Monty knows where the prize is, so when he opens the door with the booby prize, he hasn't really given you any information. The two doors still have a combined two-thirds chance of being right. However, those two-thirds odds are now all on the unopened door.

When Monty opened the door with the booby prize and lets you choose, your door still only has a one-and-three chance of having the prize. When he opened the door, the odds didn't change to one and two. If you still don't understand why you should switch doors, consider if there were more than three doors. Instead of three doors, let's say there were 100 doors. If you pick one door, you have a 1% chance of winning the prize. If Monty then opens 98 doors with a booby prize and gives you the option to pick the other door, your odds aren't 50-50.

If that were true, then every time you picked a door, you'd have a 50-50 chance, and that's clearly ridiculous with 100 doors. If you still aren't convinced, there are plenty of videos of experiments that show the answer is correct. And there are computer simulations and real-world experiments which have been run that prove it's true. And if after all of the mathematics, logic, and experimental evidence still doesn't convince you, you can find videos of Monty Hall himself confirming that the correct answer is to switch the doors. Executive producer of Everything Everywhere Daily is James Mackala. The associate producer is Thor Thompson. Remember to leave a five-star review to get your review read on the show.

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