Odds & Evens

Introduction

Look at this activity from nRich:

On the right is a set of numbered balls (2, 3, 4, 5, 6) used for a game.

To play the game, the balls are mixed up and two balls are randomly picked out together.

  • For example: one ball numbered 4 and one ball numbered 5

The numbers on the balls are added together: 4+5=9

  • If the total is even, you win.

  • If the total is odd, you lose.

Do you think the game is fair? Why/ why not?

Support

Take a look at the prompts below if you need some help:

  • You might like to experiment with this interactivity.

    • Why would we want to run it a 100 times?

    • What does the graph show you?

    • What does the table show you?

    • What is the experimental probability? Wha happens when you run it a 100 times again?

    • Does the experiment suggest it's fair or not? Why/ why not?

  • Now let's try to work it out theoretically.

    • What is a systematic way of presenting the sample space?

    • How is this different to the last lesson?

    • How do you calculate the theoretical probability of winning? What does it suggest about whether this game is fair or not?

Here are three more sets of balls:

What proportion of the time would you expect to win each game?

Which set would you choose to play with, to maximise your chances of winning?

  • Click the hyperlinks above to experiment using the interactivity.

Further Questions and Challenges

Can you find a set of balls where the chance of getting an even total is the same as the chance of getting an odd total? Either explore on your own or look at the following prompts:

  1. Create you own game using exactly 6 counters.

  • Choose the numbers on the counters to give yourself the best chance of winning.

  • Choose the numbers on the counters to give yourself the worst chance of winning.

  • Choose the numbers on the counters to make it a fair game, is that possible? If so, what are the numbers? If not, make it as close to a fair game as possible.

  1. Now repeat Q1 with 4 counters, then try 9 counters.

  2. What amounts of counters do you think might give a fair game, and which numbers would you need to use?

  • You can click on the purple cog to change the sets of numbers - just list the numbers you want to use, separated by a space

  • How many sets of balls with this property can you find?

  • What do you notice about the number of odd and even balls in your sets?

  • Can you come up with a rule?