The cube and coin challenges

Purpose

In this unit we play several games based on coloured cubes and coins. The purpose is to get some idea of the relative rate at which things happen and think about the concept of a fair game.

Specific Learning Outcomes
  • Recognise that not all things occur with the same likelihood.
  • Observe that some things are fairer than others.
  • Explore adjusting the rules of games to make them fairer.
Description of Mathematics

In this unit we look at different situations in order to try to predict whether things occur with the same probability or not. The idea is to try to achieve a range of experiences that will help the students to see what makes some things occur more often than other things. They will also play the games so that they can hone their intuition and help them test their predictions. Our lives are affected by probability of some sort ranging from insurance issues (what are the chances of having a car accident or having a burglary) to the chance of the weather affecting a wedding. It is therefore important to have some idea of what things influence events. This unit tries to do this in two types of fairly controlled events.

Required Resource Materials
  • Multi-link cubes
  • Coins
Activity

Teacher's Notes

There are several 'cube and coin challenges' in this unit. The point of these Notes is to explain the rules of the 'challenges' and the probability that underpins them. Each of the 'challenges' involves choosing a number of cubes from a bag containing several coloured cubes or tossing some coins and seeing how many heads and tails come up. The 'challenge' is won if the correct choice is made. Students record the number of 'challenges' they win and the number they lose.

Cube Challenge I:

Bag contents: one red and one blue multi-link cube
Choose one cube
To win the challenge: take a red cube.
Clearly this is a fair 'challenge' with each player equally likely to win or lose the 'challenge'.

Cube Challenge II:

Bag contents: one red and two blue multi-link cubes
Choose one cube
To win the challenge: take a red cube.

This is not a fair 'challenge'. The most likely event is that a blue cube will be taken. Choosing red is less likely. In fact players here are twice as likely to lose the 'challenge' as to win it.

If you want to change this to a fair 'challenge', then the easiest way is to add one red cube. If you want to change this to a 'challenge' that you will win, then the easiest way is to add many red cubes. The more red cubes you add, the more likely you are to win the 'challenge'.

Cube Challenge III:

Bag contents: one red, one blue and one green multi-link cube
Choose one cube
To win the challenge: take a red cube.

This is not a fair 'challenge'. There are three equally likely events: take a red; take a blue; take a green. But as far as the 'challenge' is concerned the outcomes are 'win' or 'lose'. And the 'lose' option is more likely (twice as likely) than the 'win' option.

If you want to change this to a fair 'challenge', then the easiest way is to add one red cube. If you want to change this to a 'challenge' that you will win, then the easiest way is to add many red cubes. The more red cubes you add, the more likely you are to win the 'challenge'.

Cube Challenge IV:

Bag contents: three red and two blue multi-link cubes
Choose one cube
To win the challenge: take a red cube.

Clearly this is not a fair 'challenge'. There are two unequal events: take a red; take a blue; and taking a red is more likely than taking a blue. But as far as the 'challenge' is concerned the outcomes are 'win' (= take a red) or 'lose' (= take a blue). The 'win' option is more likely to occur (three out of five times).

If you want to change this to a fair 'challenge', then the easiest way is to add one blue cube. If you want to change this to a 'challenge' that you will win more often, then the easiest way is to add many red cubes. The more red cubes you add, the more likely you are to win the 'challenge'.

Cube Challenge V:

Bag contents: four red, two blue and three green multi-link cubes
Choose one cube
To win the challenge: take a red cube.

Clearly this is not a fair 'challenge'. There are three unequal events: take a red; take a blue; and taking a red is more likely than taking a green which is more likely than taking a blue. But as far as the 'challenge' is concerned the outcomes are 'win' or 'lose'. Here 'win the challenge' = 'take a red' and 'lose the challenge' = 'take a blue or green'. The 'win' option is less likely to occur (four times out of nine).

If you want to change this to a fair 'challenge', then the easiest way is to add one red cube. If you want to change this to a 'challenge' that you will win more often, then the easiest way is to add many red cubes. The more red cubes you add, the more likely you are to win the 'challenge'.

Cube Challenge VI:

Bag contents: two red and two blue multi-link cubes
Choose two cubes from the bag
To win the challenge: take two red cubes.

This is not a fair 'challenge'. There are three unequal events: take two reds; take two blues; and take a red and a blue. The probability of two reds is the same as that of two blues and is 1/6. The probability of one of each is 2/3.

However, as far as the 'challenge' is concerned the outcomes are 'win' or 'lose'. Here 'win the challenge' = 'take two reds' and 'lose the challenge' = 'take anything else'. The 'win' option is less likely to occur (one time out of six).

It is not easy to see how to change this to a fair challenge. You could change a blue cube for a red one in the initial mix.

Coin Challenge I:

One coin
Toss the coin
To win the challenge: a head shows.

This is clearly a fair 'challenge'.

Coin Challenge II:

Two coins
Toss the coins
To win the challenge: two heads show.

This is not a fair 'challenge'. To win the challenge you need two heads but this is only as likely as getting two tails. The fact that you can also get one of each and lose the challenge means that you lose more often than you win.

To make it fair you would need to (i) have two heads or two tails show; or (ii) decide that one head and one tail, would give you the win.

Coin Challenge III:

Three coins
Toss the coins
To win the challenge: two heads and a tail show.

This is not a fair 'challenge'. To see precisely why this is so, look at Heads and Tails, Level 4.

To make it fair you would need to (i) have at least two heads show; or (ii) decide that no heads or only one head, would give you the win. There don't seem to be any other easy options.

Coin Challenge IV:

Four coins
Toss the coins
To win the challenge: three heads and a tail show.

This is not a fair 'challenge'. To see precisely why this is so, look at Heads and Tails, Level 4.

To make it fair you would need to (i) have an even number of heads show (where none is even); or (ii) decide that an odd number of heads show, would give you the win.

Coin Challenge V:

Five coins
Toss the coins
To win the challenge: four heads and a tail show.

This is not a fair 'challenge'. To see precisely why this is so, look at Heads and Tails, Level 4.

To make it fair you would need to (i) have an even number of heads show (where none is even); or (ii) decide that an odd number of heads show, would give you the win.

Teaching Sequence

Getting Started

In this session we introduce the ideas of the Cube Challenges and the Coin Challenges.

  1. Introduce the ideas of the 'challenges'.
  2. In turn play Cube Challenges I and II and Coin Challenges I and II.
  3. For each 'challenge' ask is the 'challenge' is fair or not?
  4. Play the 'challenge' several times and see if the students want to review their decision on fairness.
  5. If they are sure that it is not fair, they should make the game fair and play it again under the new rules
  6. Finally they should make a final decision on the fairness of the 'challenge' and have some reason(s) to support their fairness decision.

Exploring

In the next three sessions the students are allowed to work on the various challenges in pairs.

Here the students work in groups of two or three on Cube Challenges III, IV, V, and VI and Coin Challenges III, IV and V. You will need to make sure that they are mathematically on track and give help where it is needed. Some of the later challenges are particularly hard so they may need help on these.

Before each 'challenge' the group should

  1. Record whether they think that the 'challenge' is fair or not play the 'challenge' several times
  2. Review their earlier decision on the fairness or not of the 'challenge'
  3. If they are sure that it is not fair, they should make the game fair
  4. Play the game again under the new rules
  5. make a final decision on the fairness of the 'challenge'
  6. Have an argument to support their fairness decision.

Reflecting

In this session we help consider the relative values of different probabilities in the light of the 'challenges' that have been played.

  1. Review the 'challenges'. In particular get the class to think about the probability of all possible events (one red, one blue, one green, etc.). They should not necessarily know the actual value of the theoretical probabilities involved but should have some reason for ordering them from the most likely to the least likely.
  2. In groups, give them the opportunity to make a new 'challenge' using cubes or coins or anything else that they fancy. In each case they should have some idea of whether this is a fair 'challenge' or not.
  3. Let groups swap 'challenges' and play them.
  4. Give pairs of groups the chance to talk about their challenges and say why they are or are not fair.

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