What it shows
In the three-door puzzle, switching wins two times out of three, though many people feel that it is a fifty-fifty choice. Playing it many times and listing all the outcomes shows why.
The science
You pick one of three cups. The chance that your pick is right is 1/3, and the chance that the prize is in one of the other two is 2/3. The host, who knows where the prize is, always opens an empty cup among the other two. So the 2/3 chance does not disappear, it moves onto the one remaining cup. Staying wins with probability 1/3 and switching wins with probability 2/3. This depends on the host knowing and always opening an empty cup.
Materials
Cost: free. Many items come from home or from waste; spare parts such as LEDs, motors and wire are in the store.
- Three paper cups
- One coin or small ball to hide
- A partner to act as the host
- A tally table with two columns: stay and switch
- Paper and pencil
- A tree diagram on a chart
- A calculator
Safety
For this project: No special safety. Play fairly and do not peek.
Never use mains electricity: power models with cells, batteries or a low-voltage adapter. Sharp and hot tools, flames and lab chemicals need an adult or your teacher beside you. Keep magnets, button cells and small parts away from young children, and wash your hands after making.
All the safety rulesStep by step
8 steps. Do them in order, and test each part before you go on to the next. Write what you notice in a notebook as you go.
The host hides the coin under one of three cups without the player seeing. The host should choose the cup with a die, with 1 or 2 for the first cup, 3 or 4 for the second, and 5 or 6 for the third.
The player picks one cup by pointing to it.
The host lifts one of the other two cups which is empty. If both are empty, the host picks either.
In the first 50 games the player always stays. Record a win or a loss in the stay column.
In the next 50 games the player always switches to the other closed cup. Record in the switch column.
Work out the percentage of wins for staying and for switching and draw a bar chart of both.
Draw a tree diagram: three equally likely positions of the coin and the player always picks cup 1. List the nine outcomes to show that switching wins in two of the three cases.
Try the game with 10 cups where the host opens eight empty ones and note how the advantage of switching changes.
What you should see
About one third of the stay games and about two thirds of the switch games are won, and the tree diagram explains why. With ten cups, switching wins about 9 times in 10.
Viva questions
A teacher or a judge will ask how it works. Try to answer each question before you open it, then compare.
1Why is the answer not 50-50?
The host’s opening of an empty cup uses information, so the first choice keeps its 1/3 chance and the other cup holds 2/3.
2What if the host opened a cup at random and it happened to be empty?
Then the chances are equal, because the host gave no new information.
3Why do many trials help?
The observed frequency of winning gets closer to the true probability as the trials increase.
4What is the probability of winning by switching with 10 cups?
9/10, if the host opens 8 empty cups.
Take it further
Write a small program or a spreadsheet formula that plays 10,000 games and shows the two win rates.
Change one thing, measure what happens, and add a table or chart of your readings: that is what turns a model into your own investigation.
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