Argyle’s Math Page

A math conundrum - the Monty Hall Dilemma

Monte Halperin, OC (born August 25, 1921 in Winnipeg, Manitoba, Canada), better known by the stage name Monty Hall, is a Canadian-born emcee, producer, actor, singer and sportscaster, best known as host of the television game show Let's Make a Deal.

Hall was the host of the game show Let's Make a Deal,. Let's Make a Deal aired on NBC daytime from December 30, 1963 to December 27, 1968 and on ABC daytime from December 30, 1968 to July 9, 1976, along with two primetime runs.

The Monty Hall Dilemma was discussed in the popular "Ask Marylin" question-and-answer column of the Parade magazine. Details can also be found in the "Power of Logical Thinking" by Marylin vos Savant, St. Martin's Press, 1996.

Marylin received the following question:

Suppose you're on a game show, and you're given the choice of three doors. Behind one door is a car, behind the others, goats. You pick a door, say number 1, and the host, who knows what's behind the doors, opens another door, say number 3, which has a goat. He says to you, "Do you want to pick door number 2?" Is it to your advantage to switch your choice of doors?

Marylin's response caused an avalanche of correspondence, mostly from people who would not accept her solution. Several iterations of correspondence ensued. Eventually, she issued a call to Math teachers among her readers to organize experiments and send her the charts. Some readers with access to computers ran computer simulations. Years ago I concluded this paragraph with the sentence, At long last, the truth was established and accepted.

In fact truth proved to be more complex. With the accumulated experience, it is safe to assert that the major thrust of the controversy was not directed at Marylin's solution but at her interpretation of Craig's formulation. Indeed, there are two ways to interpret Monty's behavior as described by Craig Whitaker, "... and the host, who knows what's behind the doors, opens another door, say number 3, which has a goat."

One interpretation stems from the disregard of the piece of information conveyed by the phrase "the host, who knows what's behind the doors." According to this interpretation, Craig's statement nowhere says that the host acted on his knowledge of what's behind the doors. He just opens the door behind which (apparently by pure chance) is found a goat. The only reason to mention his knowledge one may think is to avoid the need to mention his lack of surprise at the find. During the whole operation the host remained as nonchalant as he is supposed to be.

Marylin, on the other hand, chose another interpretation. She sensed that there is a reason that host's knowledge of the location of the items behind the doors has been mentioned explicitly. And although it was not followed by an equally explicit statement to the effect that, based on his knowledge, the host always opens the door to reveal a goat, this is what the problem was about. For more on the controversy and its history please check the Monty Hall Problem article at the wikipedia.

What follows is based on Marylin's interpretation.

Below is one simulation you may try on your computer. For simplicity, I do not hide goats behind the doors. There is only one 'abstract' prize. You may either hit on the right door or miss it. You make your selection by pressing small round buttons below input controls that substitute for the doors. Down below other controls update experiment statistics even as you progress.

Approximately every second the program clears 'door' controls and is waiting for your selection. Before you start, set up a desired total number of experiments. With every selection it will decrease by 1.

Two controversial solutions are given after the puzzle. Which is the right one?

The story so far...

Comic book Math Hero of the day:When Bobby was young, his mother died in a car accident, and he was finding it hard to adapt to these new circumstances. To try and help remedy the situation, his father took him on a trip to the Pacific Islands. While taking a cruise one afternoon, they encountered a storm, and while Bobby washed up on a tropical island, his father was lost at sea.

While on the island, Bobby developed a love of books, learning, and especially math. When he was finally rescued, he proceeded to get his PhD in math, and eventually decided to fight problems and crime as "The Innovative Dr. Knowledge", shown below:

i dare you to read

all about

math at Argyle

This movie was shot by Argyle students. It is about the dangers of Credit Card Usage - Shot in Hi Definition

Loves Me, Loves Me Not (Do the Math)

An Integration of Math and Romeo & Juliet

“In the spring,” wrote Tennyson, “a young man’s fancy lightly turns to thoughts of love.” And so in keeping with the spirit of the season, this week’s column looks at love affairs — mathematically. The analysis is offered tongue in cheek, but it does touch on a serious point: that the laws of nature are written as differential equations. It also helps explain why, in the words of another poet, “the course of true love never did run smooth.”

To illustrate the approach, suppose Romeo is in love with Juliet, but in our version of the story, Juliet is a fickle lover. The more Romeo loves her, the more she wants to run away and hide. But when he takes the hint and backs off, she begins to find him strangely attractive. He, on the other hand, tends to echo her: he warms up when she loves him and cools down when she hates him.

What happens to our star-crossed lovers? How does their love ebb and flow over time? That’s where the math comes in. By writing equations that summarize how Romeo and Juliet respond to each other’s affections and then solving those equations with calculus, we can predict the course of their affair. The resulting forecast for this couple is, tragically, a never-ending cycle of love and hate. At least they manage to achieve simultaneous love a quarter of the time.

The model can be made more realistic in various ways. For instance, Romeo might react to his own feelings as well as to Juliet’s. He might be the type of guy who is so worried about throwing himself at her that he slows himself down as his love for her grows. Or he might be the other type, one who loves feeling in love so much that he loves her all the more for it.

Add to those possibilities the two ways Romeo could react to Juliet’s affections — either increasing or decreasing his own — and you see that there are four personality types, each corresponding to a different romantic style.

My students and those in Peter Christopher’s class at Worcester Polytechnic Institute have suggested such descriptive names as Hermit and Malevolent Misanthrope for the particular kind of Romeo who damps out his own love and also recoils from Juliet’s. Whereas the sort of Romeo who gets pumped by his own ardor but turned off by Juliet’s has been called a Narcissistic Nerd, Better Latent Than Never, and a Flirting Fink. (Feel free to post your own suggested names for these two types and the other two possibilities.)

Although these examples are whimsical, the equations that arise in them are of the far-reaching kind known as differential equations. They represent the most powerful tool humanity has ever created for making sense of the material world. Sir Isaac Newton used them to solve the ancient mystery of planetary motion. In so doing, he unified the heavens and the earth, showing that the same laws of motion applied to both.

In the 300 years since Newton, mankind has come to realize that the laws of physics are always expressed in the language of differential equations. This is true for the equations governing the flow of heat, air and water; for the laws of electricity and magnetism; even for the unfamiliar and often counterintuitive atomic realm where quantum mechanics reigns.

In all cases, the business of theoretical physics boils down to finding the right differential equations and solving them. When Newton discovered this key to the secrets of the universe, he felt it was so precious that he published it only as an anagram in Latin. Loosely translated, it reads: “It is useful to solve differential equations.”

The silly idea that love affairs might progress in a similar way occurred to me when I was in love for the first time, trying to understand my girlfriend’s baffling behavior. It was a summer romance at the end of my sophomore year in college. I was a lot like the first Romeo above, and she was even more like the first Juliet. The cycling of our relationship was driving me crazy until I realized that we were both acting mechanically, following simple rules of push and pull. But by the end of the summer my equations started to break down, and I was even more mystified than ever. As it turned out, the explanation was simple. There was an important variable that I’d left out of the equations — her old boyfriend wanted her back.

In mathematics we call this a three-body problem. It’s notoriously intractable, especially in the astronomical context where it first arose. After Newton solved the differential equations for the two-body problem (thus explaining why the planets move in elliptical orbits around the sun), he turned his attention to the three-body problem for the sun, earth and moon. He couldn’t solve it, and neither could anyone else. It later turned out that the three-body problem contains the seeds of chaos, rendering its behavior unpredictable in the long run.

Newton knew nothing about chaotic dynamics, but he did tell his friend Edmund Halley that the three-body problem had “made his head ache, and kept him awake so often, that he would think of it no more.”

I’m with you there, Sir Isaac.