THESE ACTIVITIES DID NOT FORM PART OF THE ORIGINAL 'MATHEMATICAL THINKING' PROJECT
GETTING STARTED
- If you are sharing or not using your own journal, Write your name at the top of the page
- In your journal or at the top of your page, write the date and the title
CARS AND GOATS
EXPLANATION FOR TEACHERS
The cars and goats problem became world famous in 1990. The author Marilyn Vos Savant, was, according to the Guiness Book of Records at the time, the person with the highest IQ in the world.
Rewriting in her own words a problem posed to her by a correspondent, Craig Whitaker, Vos Savant asked the following:
“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 No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?”
Is it to your advantage to switch your choice?”.
Video 1. Cars and Goats - Thinking Like a Mathematician (6min)
Vos Savant proceeded to give a number of simple arguments for the good answer: “switch”, it doubles your chance of winning the car.
- A stayer wins if and only if a switcher loses.
- A stayer only wins one third of the time.
- Hence a switcher only loses one third of the time, and wins two thirds of the time.
A very detailed explanation of the problem and its many variations is contained in The Encyclopedia of Mathematics
Something for a rainy day
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EXAMPLE CLASSROOM ACTIVITY
MATERIALS
- Three decorative cardboard boxes
- One toy car
- Two toy goats
PREPARATION Each of the cardboard boxes should ideally be clearly numbered (1,2 and 3), have a secure lid/door and be brightly decorated.
The boxes should be designed/positioned to ensure players cannot see items as they are placed into or stored in a box.
The car should be randomly assigned to a box for each game. For example, throw a die (1or2=box1, 3or4=box2 5or6=box3) or some other way of determining which box the car is placed in for each game).
The car:box number for each game can be determined and documented in advance.
The games should be played 30 or more times to ensure that the results are minimally reliable.
The result of each of the games should be recorded for later analysis/discussion.
THE CLASSROOM ACTIVITY
Date: __.__.18
Title: CARS AND GOATS
Each student takes a turn as a guest/player in an imaginary TV game show.
- The TV host (teacher or student experimenter/scientist) shows each guest three doors
- The guest/player is told that behind one of the doors there is a car.
- Behind the other two doors, there are goats.
The guest/player must try to win the car. To win the car:
- The TV host/experimenter asks the guest/player to choose a door.
- After the guest/player makes a choice, the TV host/experimenter opens a different door, revealing a goat.
- The TV host/experimenter then says something like “We're down to two pots now, and I'm going to give you the chance to change your mind”: To stay with the initial choice, or change their mind and switch to the remaining closed door.
THE PROBLEM
- If you were the guest/player, do you think it makes any difference which door you choose?
- If you played this game 30 times, how many times do you think a player would win a car?
Write down how many times you predict a player would win if they played 30 games and draw a circle around your prediction.
Write down your justifications/proof, so that you can share and explain it to other students.
Design an experiment that would test your prediction.
Perform your experiment and record the data (for example, in a table of results in your journal)
- Write down the number of games and draw a triangle around that number
- Write down the number of cars that were won in your experiment and draw a square around it.
Do the experimental results (data) support your prediction?
REMEMBER
- The game show host/experimenter knows in advance which of the three doors hides the car.
- Whatever door the guest/player initially chooses, the host/experimenter knows which of the remaining doors to open to reveal a goat.
- More certain still, the guest/player knows that the host/experimenter will certainly not open the door that hides the car.