Let's Make A Deal

I like this simple game because it challenges our intuitions. It shows how simple logic puzzles that involve negation and probability are not so easy for many of us.

In her September 9th, 1990 Parade Magazine "Ask Marilyn" column, Marilyn vos Savant created quite a stir when she advised on the proper course of action in a "Let's Make A Deal" type of game scenario sent in by a reader. Thousands of people - including hundreds of mathematically inclined academics - wrote in to say that her answer was wrong, and some even lambasted her for leading so many astray.

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?

Craig. F. Whitaker, Columbia, MD

Try the game out to see what you think ...

The Game
Games 0 Won 0 Lost 0
    Sim:  

1) Select a door; then host will select a goat door.
2) Now decide whether you want to select the remaining door instead. Click on your final door of choice.
3) To play again, click on any door, or press "Restart"
Try a "switch" or "hold" strategy simulation button which plays the game fifty times according to the given strategy!

Discussion

To switch or not to switch? Marilyn said yes, always switch, since that way you will win two-thirds of the time! You may find the easiest way to understand this is by taking on the role of the show host. Your contestant - Ralf, say - makes a choice, right (1/3 chance) or wrong (2/3 chance). When Ralf chooses the prize door (1/3 of the time), you are not helping him out by revealing (opening) one of the remaining goat-doors and asking him if he wants to switch to the remaining goat door (so switch = bad 1/3 of the time). But if he's wrong, 2/3d's of the time, then he chose a goat door. When you reveal the other goat-door, you are definitely helping him out by leaving only the prize door to switch to.

Debate about the correct answer mainly stems from the somewhat vague description of the host's behaviour. Is the host using his/her backstage knowledge to always expose the door (or one of two) that has a goat behind it? Most of us implicitly assume this is the rule at work. After all, the alternative - that the host randomly chooses a door regardless of what is behind it - entails that he prematurely ends the game a third of the time by exposing the prize door! This game has also been named after Monte Hall (a game show host) as "The Monte Hall Problem".

Classroom Fun

This gameshow can become a fun and tangible classroom activity for students (even into university, as the movie "21" shows in its opening scene). Explain the game to your students and have them make and paint 3-door gameshow stages and prizes out of cardboard. They can be creative about what the stage theme is - barn doors, haunted house, etc. - and what the prizes are - 2 goats & car / 2 skunks & lollipop / 2 zombies & 1 wizard etc. Have the stages built large enough so the game host can place prize and bane behind the doors without being seen. Include a scoreboard of some kind or have this as a separate notepad.

Then have the students break into groups of 2 or 3 to play, each taking turns for a while as the game show host, the others as player/scorekeeper. The host should randomly place the prize each time of course - a 6 sided die can be pressed into action for this, with 1,2 = door 1 ; 3,4 = door 2; 5,6 = door 3. (Without this randomization, a player may spot a pattern in the host's choice, for example if host moves the prize to a new door on each new game - and that would compromise the results of our little probability experiment!)

This way players get the feel for how a particular strategy works long-term. Do the game hosts learn more quickly (than the players) that switching is the best strategy?

The score page can have 2 columns: Strategy (H:hold/S:switch) Win (Yes / No), with each row posting the result of a game. Classes who are doing graphing in math can then total and graph the results of hold-wins vs switch-wins for a given number of games played.