The Key Problem

The Problem

In an imaginary contest, seven people have the chance to win a car. Out of seven keys, one will start the car. The players take turns, each selecting a key and trying it in the car. When a player chooses the correct key, he or she wins the car and the contest is over.

Does each player have an equal chance of winning? Isn't it unfair that some players might not even get to choose?

Note: This is only a demonstration. We are not giving away a car.

The Analysis

Each player has an equal chance (one in seven) of winning. One way to look at this is to consider two types of risk each player faces:

These risks balance each other. Early players have high performance risk and low opportunity risk. Later players have high opportunity risk and low performance risk because some keys have been eliminated before they choose.

For example, the first player has a 100% chance of getting to choose, but only one chance in seven (about 14.29%) of choosing the correct key. The second player has six chances in seven (about 86.71%) of getting to choose, but then has one chance in six (about 16.67%) of choosing the correct key. The last player has only one chance in seven (about 14.29%) of getting to choose, but then has a 100% chance of choosing the right key. (There will be only one key left at that point.)

When you play the game, the specific opportunity risk and performance risk for each player are displayed in the expected results.

Playing the Game

You get to play the parts of all seven players. Seven keys are initially displayed across the top of the window. To choose a key, click on it. If it is the wrong key, it disappears and next player gets a turn. Continue choosing from the remaining keys. When you select the winning key, a message is displayed. When you acknowledge the message, the updated results are displayed in the lower part of the window. You can choose to see just results for the current session or to see historical results (since about June 1, 2003).

To automatically play the game multiple times, enter a number in the field to the right of the keys and then click the Auto Play button. The program automatically plays the number of iterations you specified and updates the displayed results. If you enter an especially high number, you might experience some delay as the games are processed.

Back to Between Waters