MONT 106N – Identifying Patterns Group Discussion – Chance November 18, 2009

Background

Today, we want to think about and experiment with a situation involving chances, modeled on a once-popular old-time TV show called Let’s Make a Deal. Here’s the situation:

• The game show host, a slightly sleazy huckster type named Monty Hall, shows the contestant three closed doors. He says that a valuable prize (typically a car) is behind one of them, while the other two doors conceal worthless joke “prizes” (typically live goats(!)).

• The contestant chooses one of the doors, but the door is not opened to reveal the prize behind it at that point.

• Monty, who knows where the valuable prize is, opens one of the doors concealing one of the worthless prizes.

• At this point, Monty gives the contestant the chance to either stay with the door they chose originally, or else switch to the other unopened door.

• After the contestant makes his or her choice, that door is opened and the prize is revealed.

The question is: Is it a better strategy for the contestant to stay with their original choice of door? Or is it a better strategy for them to switch to the other unopened door? Or perhaps might it be best to mix up the two possibilities, maybe by alternating between staying and switching?

Discussion Questions

A. Do you have an intuitive feeling what the best strategy should be for the contestant? If so, what do you think? Or should it matter which way the contestant makes his or her choice?

To get some data to analyze, we are going to simulate the Let’s Make a Deal game with slips of paper. In your group, designate one person as “Monty” for each of the following parts. Your group will receive three slips of paper, one marked “car” on one side, and two marked “goat” on one side. “Monty” will then place the three slips of paper in some random order (that he or she remembers!) in front of a second team member who will be the contestant. The contestant will select one slip, without turning it over. At least one of the unchosen slips is a “goat” at this point. “Monty” will then turn over one of the “goats” and the contestant will choose to hold onto the first slip or to switch, as below. Then that slip will be turned over and the third person will record whether the game was a win for the contestant (he or she got the “car”), or a loss (he or she got the other “goat”). Rotate the roles of “Monty,” the contestant, and the record keeper to keep everyone interested!

B. Play 25 rounds of the game, having the contestant always hold onto their original card when offered the opportunity to switch, and count the number of “wins.”

C. Play 25 rounds of the game, having the contestant always switch to the other unknown card when offered the opportunity to switch, and count the number of “wins.”

D. Play 25 rounds of the game, having the contestant alternate between holding onto their original card and switching when offered the opportunity to switch, and count the number of “wins.”

E. Which strategy gave the most wins for the contestant? Was this a surprise?

Assignment

Group writeups due at the end of class.