Sarah's Game

Introduction

In today's game:

• Put your counter on "start"

• Throw a fair 6 -sided die

  • If you throw 1, 2, or 3 then move one square up.

  • If you throw 4, 5 or 6 then move one space to the right.

• Predict where you think the counter will end up then play the game!

• Repeat 20 times, make sure you record the experimental probabilities of landing on each letter.

• Do you think it is equally likely to land on any of the four letters? Why/ why not? Which letter(s) are more likely to land on? Less likely?

• What are the different ways of getting to each letter?

• Hence what are the theoretical probabilities of landing on each letter?

• How are the theoretical and experimental probabilities similar/different?

• If you repeat this game 100 times, on average, how many of each letter do you expect?

Further Questions and Challenges

Experiment with different sized boards e.g. one with 5 letters like the one on the right:

• Repeat 20 times, record the experimental probabilities of landing on each letter.

• Is there a systematic way of writing down all the paths? Can you see a pattern in the number of ways?

• What’s the total number of outcomes for the size 5 board?

• What are the theoretical probabilities of landing on each letter?

• If you repeat this game 100 times, on average, how many of each letter do you expect?

• What if we have a size 6 board? A size 10 board? A size n board? How many outcomes will there be? Hint: you may wish to start with a size 3 board and work your way up.

• What’s the best way to record our results?

• Let's change the game a little:

• If you throw 1, 2, 3 or 4 then move one square up.

• If you throw 5 or 6 then move one space to the right.

• With a size 2 game board (see the right), which letter do you think the counter will end up?

  • How would you record the results?

  • What would the experimental probabilities be?

  • What would the theoretical probabilities be?

  • How are the theoretical and experimental probabilities similar/different?

• Now repeat the above with a size 3 game board

  • How would you record the results?

  • What would the experimental probabilities be?

  • What would the theoretical probabilities be?

  • How are the theoretical and experimental probabilities similar/different?

  • How is this similar to what you learnt in the "Spinners" lesson?

• Can you extend this to a size 4 game board? size 5 etc?