Grade 7 | Unit 7.4 | Page 14

Unit 7.4Completed Teacher’s Learning Activity Sheet

Chance and probability – coins and dice

(The answers to the questions are in red.)

You may ask, “What is the probability of my correctly predicting the outcome, and thus of winning?”

In your group, do the following exercises to find out the probability of winning or losing.

(If there is not enough time in class to complete the activity, learners can complete it at home for homework. A copy ofthe answer sheet can be handed out to learners once they have completed the task.)

Heads or tails?

(Coins have a “head” side – the side with the South African emblem or coat of arms – and a “tail” side – the side withthe amount written on it and the picture of a flower or animal.)

1. Use one coin for the following exercise. Toss a coin in the air. Write down the result on the line below. Now, tossit 5 more times and after each toss, write down the result. (H = “heads”; T = “tails”)

_________ __________ ___________ ___________ __________ __________

(Learners fill in the actual results. Since these will differ from group to group, there is no need to go over the resultsfor questions 1-4. The reason for their inclusion is to show learners that there is no certainty in being able topredict the correct outcome. Ask learners: What did you find when you compared your guesses with the actualoutcome? Was it because you were skilful or merely lucky that you guessed correctly?)

2. Before tossing the coin again, each person in the group must guess beforehand whether a “head” or “tail” willcome up in the next three throws. Write your own three guesses in the three lines below (H = “heads”; T = “tails”):

Your guess _______ _______ _______

Now, once you’ve written down your predictions, a person in the group must toss the coin. Write down the actualside that comes up in the three throws.

Actual result _______ _______ _______

3. Look at the previous results above. Now, once again, guess the results of the next three tosses. Write down yourown guess on the lines below:

Your guesses _______ _______ _______

Now, once you’ve written down your predictions, a person in the group must toss the coin. Write down the actualside that comes up in the three throws.

Actual result _______ _______ _______

4. How many people in your group guessed correctly for both rounds?

5. There are 2 sides to a coin, therefore there are only two possible outcomes: heads or tails:

Heads Tails

Play a game in which you can bet only one outcome (Heads or Tails). Write down which outcome you wish to bet on:

Your chances of winning are therefore: 1 in 2; or 1/2 , or 50%(Hint: divide your 1 predicted outcome by the number of possible outcomes.)Say: You can see that you cannot know for certain what the outcome will be.

6. Note why you think coin tossing is a game of skill (the more skilful player is better able to determine the outcome)or a game of chance (it’s all just luck)? ____________________________________________________________

Say: Since you cannot know for certain what the outcome will be, coin tossing is a game of chance. Also, nomatter how much you play, you don’t become more skilful at being able to predict the correct outcome.

Grade 7 | Unit 7.4 | Page 15

Dice throwing

As a group, complete the following:

1. Use one die (one die, two dice) for the following exercise. Before rolling the die, each person in the group mustguess beforehand what three numbers will come up in the next three throws. Write your own three guesses in thethree lines below (any number from 1 to 6):

Your guess _______ _______ _______

Now, once you’ve written down your predictions, a person in the group must throw the die. Write down the actualnumber that comes up in the three throws.

Actual result _______ _______ _______

How many of your guesses were correct? _________________________

2. Now repeat the game but this time, instead of one die, use two dice. Before rolling the dice, guess the sum ofthe numbers that you think will come up in the next three throws. Write your three guesses in the three lines below(This will be any number from 2 to 12).

Your guess _______ _______ _______

Once you’ve written down your predictions, a person in the group must throw the dice. Write down the actualsum of numbers that comes up in the three throws.

Actual result _______ _______ _______

How many of your guesses were correct? ______________

Ask learners to look at whether predicted outcomes tended to differ from actual outcomes. If they differed, whydo learners think they did? If they were the same, why do learners think they were the same? Was it a matter ofskill or more of luck?

3. In dice throwing, using one die, there are 6 possible outcomes. What are they?

But you can choose only one option.1 chance in _ 6; or 1/6; or 16.6%

(Hint: you can choose only 1 number, but there are 6 numbers on a die, therefore 6 possible choices.)

4. Here are all the possible ways in which a pair of dice can fall:

Die 1

Die 2

How many possible ways can the dice fall in total: 36 (count the number of blocks)

(Or put differently, how many possible outcomes are there in total?)

1 2 3 4 5 6

1:1 1:2 1:3 1:4 1:5 1:6

2:1 2:2 2:3 2:4 2:5 2:6

3:1 3:2 3:3 3:4 3:5 3:6

4:1 4:2 4:3 4:4 4:5 4:6

5:1 5:2 5:3 5:4 5:5 5:6

6:1 6:2 6:3 6:4 6:5 6:6

Grade 7 | Unit 7.4 | Page 16

5. Write down all the possible ways (combination of two numbers) in which you can roll a sum of 7, using a pair of dice:

(Hint: look at the table above and count all the various ways in which two numbers can add up to 7.)

(1:6), (2:5), (3:4), (4:3), (5:2), (6:1)

There are therefore 6 different ways in which you can get a sum of 7 with 2 dice.

6. What is the probability of your throwing a sum of 7 from 2 dice?

6 in 36; or 6/36; or 1 in 6; or 1/6; or 16.6%

(Hint: Divide the total number of possible ways the numbers from two dice can fall so that they add up to 7 (6)by the total possible number of ways that the two dice can fall to give any sum at all (36).)

7. Write down all the possible ways (combination of two numbers) in which you can roll a total of 10, using apair of dice:

_______ _______ _______

(4:6) or (5:5) or (6:4)

There are therefore 3 different ways in which 2 die can add up to 10.

(Hint: look at the table above and count all the various ways in which two numbers can add up to 10. Thereare 3 different ways of throwing a sum of 10.)

8. What is the probability of your throwing a total of 10 from two dice? _ 3 in 36; or 3/36; or 1 in 12; or 1/12; or 8.33%(Hint: Divide the total number of possible ways the numbers from two dice can fall so that they add up to 10(3) by the total possible number of ways that the two dice can fall to give any sum at all (36).)

9. Let’s imagine that you bet on a “10”. Now roll the dice. Write down the actual outcome (the sum of the two dice):

10. Let’s continue to imagine that you bet again on a “10”. But, now blow hard on the dice before throwing themagain. Write down the actual outcome:

11. You bet again on a “10”. But now, roll the dice softly when you throw them. Write down the actual outcome:

12. Now, based on the previous two outcomes, make a prediction about what the next sum will be whenyou roll the dice.

Your prediction:

Actual outcome:

Say: the exercise above shows that no matter what you do, whether you blow hard, or roll softly or base youguess on previous outcomes, it doesn’t increase your chances of predicting correctly.

13. Write down all the possible combinations in which you can roll a sum of 12, using a pair of dice: _ (6:6)There is only 1 way in which a 12 can fall.

14. What is the probability of your throwing a 12 with two dice? _ 1 in 36; or 1/36; or 2.7%

15. You bet on a “12”. Now roll the dice. Write down the actual outcome (the sum of the two dice):

Grade 7 | Unit 7.4 | Page 17

16. You bet again on a “12”. But, before you roll the dice, concentrate hard on a 12, visualize the number 12 – now roll the dice. Write down the actual outcome:

17. You bet again on a “12”. But now, let the person who is feeling lucky throw the dice. Write downthe actual outcome:

18. Now, try 5 more times to get a “12”.

Actual outcome 1:

Actual outcome 2:

Actual outcome 3:

Actual outcome 4:

Actual outcome 5:

19. Do you think you can control the outcome of a game of dice? _ You cannot control the outcome in a game ofchance. We’ll learn more about why this is so in Grade 8.

Can you predict with certainty what the outcome will be?

No, not with certainty, but when throwing two dice, some sums occur more often than others, that is, they havea higher probability of occurring. This is because there are more ways that the two dice can fall to sum tocertain numbers than others. So, for example, there are six ways of getting a sum of 7, but only one way ofgetting a sum of 12..

Are there some numbers that have a higher probability of appearing (i.e. a higher number of different ways inwhich the number can fall) than others?

Yes, a sum of 7 with two dice has a higher probability of appearing than a sum of 10; and a sum of 10 againhas a higher probability of appearing than a sum of 12.

Again, you might guess correctly, but you are much more likely to guess incorrectly.

Is dice throwing a game of chance or a game of skill? Why?

It’s a game of chance. We will see in Grade 8 why dice is a game of chance. In Grade 8 we will learn about“independent events” and “random numbers”, but for now it is enough for Grade 7 learners to say that dice isa game of chance because there is very little probability of correctly predicting the right outcome.

IT IS THEREFORE NOT SURPRISING THAT JOE, OUR COMIC BOOK CHARACTER, LOST MUCH MORE THANHE WON!

THERE IS NOTHING YOU CAN DO OR THINK OR BELIEVE THAT WILL BRING

ABOUT YOUR DESIRED RESULT. DICE IS A GAME OF PURE CHANCE!! AS ARE

SLOT MACHINES AND COIN TOSSING