Wallowing whales

Game

Students can play this game with little initial support. They will soon notice that certain numbers come up with greater frequency than others and will alter their strategy to reflect this. When they have played the game for 20–30 minutes, stop the play and discuss with your students what they have discovered.

Once they are aware that there may be better strategies, they can examine the theoretical probabilities behind the game.

Activity

Note that this activity should be set before "Dodgy Dice" (page 22 of the students’ book), which requires a more formal understanding of the same ideas.

Question 2a asks the students to complete a difference table. Along with tree diagrams, tables are a useful tool for helping them to see that, in certain situations, some outcomes are more likely than others. Tables allow students to see at a glance how many different ways there are of getting each outcome. The more ways there are, the greater the probability is. Because tables are 2-D, they have the limitation that they can only be used for 2-step events (for example, a dice rolled twice).

Question 2b asks for a bar graph showing the frequency of each of the differences. If your students are using computers for this task, they may have difficulty getting the computer to label the horizontal axis correctly. If this is the case, they should follow these steps:

• Choose Chart from the menu bar.
• Select Source Data.
• Click on the Series tab.
• Click the cursor in the panel that says “Category (X) axis labels”.
• Go to the spreadsheet and highlight the cells with the correct labels in them (0–5).
• Click on OK.

Question 2c asks the students to turn the frequencies they have found in question 2b into probabilities. Suggest that they think in terms of “chances out of 36” and then write their answers as fractions. For more discussion on this, see the notes for What’s the Chance? (page 23 of the students’ book).

Question 3b is another good opportunity for developing the concept of long-run relative frequency. Discuss with your students how they might “prove” their strategy is better. Some will want to prove it by playing the game according to that strategy. In this case, ask them,

• What will it prove if you win a game?
• Does it mean you will also win the next game?
• Will you win every game you play using this strategy? If not, how often are you likely to win?
• How could you work this out?

Students may want to play a lot of games to see what happens. At some point you can ask,

• Is there a quicker way of working out how often you should win?

This may encourage students to think about the underlying probabilities.

Game

A game for investigating probability.

Activity

1. 

Answers will vary.

2.

a.

		Dice one
		1	2	3	4	5	6
Dice two	1	0	1	2	3	4	5
	2	1	0	1	2	3	4
	3	2	1	0	1	2	3
	4	3	2	1	0	1	2
	5	4	3	2	1	0	1
	6	5	4	3	2	1	0

b. 

c.

Difference	0	1	2	3	4	5
Probability	6/36	10/36	8/36	6/36	4/36	2/36

3.

a. Finn’s strategy is unlikely to be effective. Although 1 has the highest probability, in the long run, he can expect to get it only 10 times out of 36. For the other 26 out of 36 times, his turn will count for nothing.

b. A more effective strategy is to spread his counters from 0–3, with proportionally more on 1. He then has a 30/36 probability of getting a useful result each time he rolls the dice.