*Probability progression, 3rd step *

## The purpose of the activity

In this activity, the learners further develop their concept of chance by discussing the likelihood of different events. They are introduced to ways of identifying all possible outcomes of a simple event (the event or sample space) and use fractions to assign likelihoods to outcomes.

## The teaching points

- The probability that a future event will occur can be described along a continuum from impossible to certain.
- The probability of an event is a measure of the chance of an event occurring. The probability of an event is a number between 0 and 1 and can be expressed as a fraction, as a percentage (0–100 percent), or as odds.
- A sample space is the set of all possible outcomes for an event. For example, there are 6 possible outcomes for rolling a standard six-sided dice and 36 (6 x 6) possible outcomes for rolling two standard six-sided dice.
- The outcomes for events are not usually equally likely. For example, the possible outcomes for a basketball free throw are either to make the goal or to miss it, with the likelihood of making it dependent on the skill of the player. On the other hand, tossing a fair coin does have two equally likely outcomes.
- There are two ways to measure chance. One way is to analyse the situation logically (theoretical probability), and the other way is to generate data to analyse the situation (experimental probability). Examples of situations that can be analysed theoretically include rolling dice, throwing coins and Lotto. Examples of situations that need to be analysed experimentally include the likelihood of there being an earthquake or a car accident.

## Resources

- A die for each learner or pair of learners.
- A large die for demonstration (if available). The guided teaching and learning sequence

1. Begin the session by showing the learners a die and asking them which number they think will come up if you roll it.

“What number do you think I will roll?”

“Why do you think that?”

2. Check that the learners’ responses show that they understand that it isn’t possible to predict what will happen when you roll a standard sixsided die.

3. Roll the dice and see if the learners guessed correctly.

4. Ask the learners to state the possible outcomes for rolling the dice. List these on the board.

*What are the possible numbers that I can roll? *

5. Tell the learners that this list of all the possible outcomes is called the sample or the event space: {1, 2, 3, 4, 5, 6}

6. Draw a line on the board, labelling one end “Impossible” and other “Certain”. Ask the learners if they know which numbers should go with the words. (0 and 1 or 0 and 100%.)

7. Record these on the line.

8. Ask:

“What is the chance of rolling a 3?” (1/6) Ensure the learners understand that there is 1 chance in 6 of rolling a 3 and that this can be expressed as one-sixth or 1/6 .

9. Ask a learner to put a mark on the line at 1/6 . Check that the learners understand that this mark is 1/6 of the way along the line. Also ask the learners to suggest words that could be used to describe the chance of rolling a 3.

10. Ask:

“What is the chance of getting an even number when you roll the dice?” (1/2 or 3/6 or 50%).

11. Ask the learners to explain their reasoning. Check that they understand that there are two possible outcomes in the sample space {even numbers, odd numbers} and that the two outcomes are equally likely because three of the numbers are odd (1, 3, 5) and three are even (2, 4, 6). Check also that the learners understand that 3/6 is equivalent to 1/2 and 50%.

12. Record 1/2 on the line and ask the learners to suggest words to describe this probability, for example, “equally likely” or “even chance”.

13. Ask:

“What is the chance of getting a number greater than 2?” (4/6 or 2/3)

14. Ask the learners to explain their reasoning. Check the learners understand that there are two outcomes in the sample space. One is numbers greater than 2 (3, 4, 5, 6) and the other is numbers 2 or less (1, 2).

15. Ask the learners to suggest events for rolling the dice that are:

- certain (for example, rolling a number less than 7)
- impossible (for example, rolling a number greater than 6).

## Follow-up activity

Give the learners a copy of the following diagram.

Explain that they are to:

1. Shade the counters shown in the containers so the chance of drawing a shaded counter matches the probability marker on the line.

2. Label the markers with the fraction that represents the probability of drawing a shaded counter.

3. Write words with each marker to describe the probability.

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