Theoretical probability (PDF, 53 KB)
Probability, 4th-5th steps
The purpose of the activity
In this activity, the learners develop their understanding of the difference between theoretical and experimental probabilities.
The teaching points
- 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, a percentage (0–100%), 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 single standard six-sided die and 36 (6 x 6) possible outcomes for rolling two standard six-sided dice.
- 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.
- A series of three different-coloured counters.
- A non-transparent bag.
The guided teaching and learning sequence
1. With the learners watching, put 10 coloured counters in a bag: 5 red, 3 blue and 2 yellow. Record the colours and numbers of counters placed in the bag on the board for the learners to refer to.
2. Ask the learners to state what will happen if a counter is selected at random from the bag:
“What colour counter will be selected?”
“What colours of counter could be selected?”
“What colour is most likely?” “Why?”
Some learners are likely to try to tell you that a red counter will be chosen. It is important for them to realise that they cannot be certain what colour will be selected. If one is chosen at random, it could be any of the three colours. Red is more likely than blue or yellow because there are more red counters than blue or yellow counters in the bag.
3. Now ask the learners to describe the chance of selecting a counter of each colour.
“What is the chance of selecting a red, blue or yellow counter?” “How did you work that out?”
The learners should be able to state:
- There is a 50% or 1/2 chance of selecting a red counter because 5 of the 10 counters are red.
- There is a 30% or 3/10 chance of selecting a blue counter because 3 of the 10 counters are blue.
- There is a 20% or 1/5 chance of selecting a yellow counter because 2 of the 10 counters are yellow.
4. Explain to the learners that these are the theoretical probabilities of the event.
5. Discuss what these theoretical probabilities mean in terms of selecting counters. For example, you are twice as likely to select a red counter as any other colour.
6. Ask the learners what they think will happen if you select 10 counters, one at a time from the bag, returning the counter to the bag between selections. The learners are likely to suggest that there will be 5 red counters, 3 blue counters and 2 yellow counters selected. While this is possible, it is not the only possible outcome, in fact the probability of selecting exactly 5 red, 3 blue and 2 yellow counters is quite low.
7. Ask 10 learners to each come up, select a counter, record its colour and replace it in the bag. Graph the results on the board and discuss the results.
8. Repeat several times.
9. Ask the learners to discuss in small groups why the results do not match the theoretical probabilities.
10. Share the ideas as a class. Hopefully some learners will recognise that theoretical probability is only a way of predicting what may occur, it does not tell you what will occur. Ensure that all the learners understand this.
Ask the learners to work in pairs to calculate the theoretical probabilities for tossing two coins. Then ask them to toss two coins 100 times and compare the results of the experiment with their theoretical probabilities.
- 2 Heads: 1/4 or 25%.
- 2 Tails: 1/4 or 25%.
- 1 Head, 1 Tail: 1/2 , 50%.
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