Algebra 1 Summer Institute 2014

The Stick, Flip, Switch Corollary

During a certain game show, contestants are shown three closed doors. One of the doors has a big prize behind it, and the other two have junk behind them. The contestants are asked to pick a door, which remains closed to them. Then the game show host, Monty, opens one of the other two doors and reveals the contents to the contestant. Monty always chooses a door with a gag gift behind it. The contestants are then given the option to stick with their original choice or to switch to the other unopened door.

Guess

In groups, discuss what you might do and why.

Experiment

As a class, discuss the following strategies. If you were the contestant, which of the following strategies would you choose, and why?
Strategy 1 (stick): Stick with the original door
Strategy 2 (flip): Flip a coin, stick if it shows heads, switch if it shows tails
Strategy 3 (switch): Switch to the other door

In groups of two perform the experiments. Each group should receive three cups, three prizes (1 car, 2 spiders), and one coin. Number the cups 1, 2, and 3 and take turns playing the game with one person as the host and the other as the contestant and switching roles periodically. Record, in a tables below, the number of wins and losses when the contestant switches doors and (separately) the number of wins and losses when the contestant doesn’t switch, as well as when they flip the coin. Try each strategy 12 times. After you are finished, compile the class data and compute the winning percentages for the different strategies.

Strategy 1- Stick

Player 1
Frequency / Player 2
Frequency
Win
Lose

Strategy 2 – Flip

Player 1
Frequency / Player 2
Frequency
Win
Lose

Strategy 3 – Switch

Player 1
Frequency / Player 2
Frequency
Win
Lose

Argue whether or not you should switch doors based on your opinion, research, and experiment.

Simulation

Let's walk through a simulation of each strategy.

The Stick Strategy

  1. Suppose that the spinner lands on door B. What does Monty do? (Shows you door C) What do you do? (Stick) Do you win or lose?
  2. Suppose that the spinner lands on door C. What does Monty do? (Shows you door B) What do you do? (Stick) Do you win or lose?
  3. Suppose that the spinner lands on door A. In this instance Monty shows you either door B or door C. What do you do? (Stick) Do you win or lose?

The Flip Strategy

  1. Suppose that the spinner lands on door B. Then Monty opens door C. You then flip a coin to decide whether to stick with door B or switch to door A. What is your chance of winning the prize?
  2. Suppose that the spinner lands on door C. Monty opens door B. You flip to decide between A and C. Again, what is your chance of winning the prize?
  3. Suppose that the spinner lands on door A. Monty opens B or C. You flip to decide between A and the other door. Once more, what is your chance of winning the prize?

The Switch Strategy

  1. Suppose that the spinner lands on door B. Monty opens door C. You switch to the unopened door. Do you win or lose?
  2. Suppose that the spinner lands on door C. Monty opens door B. You switch. Do you win or lose?
  3. Suppose that the spinner lands on door A. Monty shows you door B or C. You switch. Do you win or lose?

In the Excel simulation we will only use strategies 1 and 3 (stick and switch).

  1. We are going to label 6 columns the following way:
  2. Column B – Door with Grand Price
  3. Column C – Door Chosen
  4. Column D – Door Opened
  5. Column E – Switch to Door
  6. Column F – Win if Switch
  7. Column G - Win if no Switch
  1. In Cell B4 we will randomly select a door for the grand prize, type “=randbetween(1,3)”
  2. In Cell C4 we will randomly select a door, also type “=randbetween(1,3)”
  3. In Cell D4, the host will open a door different from the door with the grand price and different from the door selected by the contestant. This requires and embedded if statement. The format of a single if statement is:
  4. =If(logical_test,[value if true], [value if false])”. Copy the following formula carefully.

=IF(C4=B4,IF(B4=1,IF(RAND()<0.5, 2,3),IF(B4=3,IF(RAND()<0.5, 1,2),IF(RAND()<0.5,1,3))),IF(C4=1,IF(B4=2,3,2),IF(C4=2,IF(B4=1,3,1),IF(B4=2,1,2))))

  1. In Cell E4 we will indicate the number of the door if we switch with another if statement:
  1. =IF(D4=1,IF(C4=2,3,2),IF(D4=2,IF(C4=1,3,1),IF(C4=1,2,1)))
  1. In Cell F4 we will show if by switching we win the grand prize with the number 1:
  1. =IF(E4=B4,1,0)
  1. In Cell G4 we will show if by not switching, we win the grand prize
  1. =IF(B2=C4,1,0)
  1. We can now drag down these formulas as far as you like (10000 cells down)
  1. We need to count the number of times we win under each strategy by adding the ones in columns F and G and calculate the percentage.
  2. In column H3 type “=Sum(F4:F10003)”
  3. In column I3 type “=100*AVERAGE(F4:F10003)”
  4. In column J3 type “=Sum(G4:G10003)”
  5. In column K3 type “=100*AVERAGE(G4:G10003)”

Are the experimental percentages close to the ones you and your partner got when doing the experiment part of this activity?

Theoretical Model

What is the reason for the outcome? The big questions in Monty's Dilemma are (1) What do you know and (2) When do you know it?

In your group, make a careful analysis of simulations using tree diagrams.

Assume that the prize is behind door A. There is a 1/3 probability of (randomly) choosing each door. Monty must reveal a gag gift behind a remaining door. In the event we chose door A initially (the one with the prize), Monty can show us either B or C. We assume that the probability is1/2 that Monty will show us B and1/2 that he will show us C. If we chose B, Monty must reveal door C and leave A. If we chose C, Monty must reveal door B and leave A. Thus, initially, we obtain the tree diagram in the figure below.

Complete this diagram in three different ways, corresponding to each of the three strategies, stick, flip, or switch. The stick and switch strategies determine immediately which door is finally chosen along each of the branches. The flip tree requires adding two more branches at every node, each with a probability of 1/2 that we either keep our original choice or take Monty's offer to switch. Again, without loss of generality, we assume that the prize is behind door A.

After completing the tree diagrams, complete the following table with the mathematical probabilities:

Strategy / Door A / Door B / Door C
Stick
Flip
Switch

1