*Probability progression, 4th-5th steps *

## The purpose of the activity

In this activity, the learners use the frequencies of outcomes to predict the likelihood that an event will occur. The activity reinforces the idea that probabilities are not absolute predictors in the short run.

## 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 or a percentage (0–100 percent).
- A sample space is the set of all possible outcomes for an event. For example, there are six possible outcomes for rolling a single standard six-sided die.
- 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).
- Discuss with the learners examples of situations that can be analysed theoretically, including rolling dice, throwing coins and Lotto, and examples of situations that need to be analysed experimentally, including the likelihood of there being an earthquake or a car accident.

## Resources

- Small plastic cups.
- A saucer.
- A coin.

## The guided teaching and learning sequence

1. Tell the learners that you are going to toss a fair coin. Ask those who think it is going to land as heads to stand up. Assuming that there are roughly equal numbers standing and sitting, ask: “Why do you think about half of you are standing and the others sitting?” (Check that the learners know there are two outcomes and that they are equally likely.)

2. Toss the coin, noting the outcome.

3. Ask:

“If I tossed the coin one hundred times, how many heads do you think I would get?” “Why do you think that?”

“Do you think that I will get exactly 50?”

“Why or why not?”

(You can expect about 50 heads and 50 tails, although anything in the range of 40–60 is reasonable.)

4. Next show the learners a small plastic cup and tell them that you are going to toss it in the air and let it land on the ground. Ask the learners to tell you the possible outcomes for the cup landing. Record these on the board (upside down, right way up, on side).

5. Remind the learners that this list of outcomes is called the sample space.

“Do you think that the 3 outcomes are equally likely?” “Why or why not?”

“Which of these outcomes do you think is most likely?”

“Can you estimate the probability of the cup landing the right way up?”

6. Ensure that the learners understand it isn’t possible to analyse the cup-tossing event mathematically and that you need to conduct an experiment to estimate the probabilities of the three possible outcomes.

7. Have the learners work in pairs and give each pair a small plastic cup and tell them that they are going to toss it 20 times and record how it lands each time. Before they begin tossing the cup, encourage the learners to agree on a uniform method for tossing the cup (for example, standing up, flipping the cups to the same height, letting the cups land on the floor or on a table).

8. Ask the learners to share the results of their 20 trials. Discuss the differences and generate reasons for them. Possible reasons include: samples (cups’ size and shape) may vary, cup-tossing techniques may vary.

9. Ask the learners how they could use the data to estimate the probability for the three cuptossing outcomes. In the discussion, work towards ensuring the learners understand that a more accurate prediction can be made if the results of the class’s trials are combined.

10. Combine the class’s trials and use this data

to give a better estimate of the probability of the three outcomes, using fractions and percentages.

Check the learners understand why the fractions (and percentages) when added together equal 1 (or 100%).

Note: This activity also gives you the opportunity to ensure the learners understand how to calculate percentages from fractions.

11. In summary, draw a line labelled at either end 0 and 1 respectively (100%). Ask for volunteers to locate the three outcomes on the line and to give a language-based description of each outcome.

This activity also provides an opportunity to check the learners understand the relative size of various fractions and percentages and whether they understand which events are very unlikely or moderately or highly likely.

12. Discuss with the learners real-world situations where the probabilities need to be calculated from frequencies (insurance company premiums, weather forecasting, the chance of getting breast cancer, etc). Ensure the learners understand that one limitation of the relative frequency approach is that the probabilities are only estimates because they are based on finite samples of collected data.

## Follow-up activity

Give each learner (or pair of learners) two counters and a saucer. Ask the learners to calculate the probability of flicking a counter into the saucer from a distance of 10 centimetres. As this is a skills-related activity, each learner’s probability of success will be different.

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