Combined Probability

Further Ideas

A simple game can be used to help students develop their understanding of combined probabilities. The rules are as follows:

• Take turns to throw a die.
• On your turn you can throw the die as many times as you like.
• When you stop throwing, the total of the numbers thrown is added to your score.
• If you throw a one before you decide to stop, your turn is over and you score nothing.
• The first player to reach 100 is the winner. You do not have to score exactly 100 to win.

Students should play the game several times and consider the strategies that they think will be advantageous. The group can compare and discuss these, allowing exploration of concepts of luck and independence of the throws. As they get more familiar with the game they can investigate the mathematical probabilities of combinations of throws.

Students can consider questions like:

• What is the probability of throwing a one on your first throw? What is the probability in subsequent throws?
• What is the probability of not throwing a one on your first throw? On your second, third, fourth, fifth throws?
• What are the shortest sequences of throws that would win the game in a single turn? What is the probability of throwing one of these sequences?
• If you have a score of 94, what is the probability that you will win on your next throw? After two throws?... What if your score is 97?...
• Is the game different if you score zero when you throw a six rather than a one?
• What is the probability that no one scores on their first throw if there are 2, 3, 4 players...?
• If there are 90 players, how many of them would you expect to score nothing on their first throw?

You could use tree diagrams (showing, for example, outcomes of one and not-one) to illustrate the results of successive throws.