*Probability progression, 6th step *

## The purpose of the activity

In this activity, the learners explore issues related to gambling to further develop their concept of chance. They learn to use tree diagrams, organised lists and to multiply probabilities to calculate the theoretical probability of winning Lotto.

## 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.
- 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.
- Discuss with the learners the fact that theoretical probability is only a way of predicting what
*may *occur, and it does not tell you what *will *occur.

## Resources

## The guided teaching and learning sequence

1. Tell the learners that you are going to play a simple Lotto game that involves just three numbers (1, 2, 3) and that the game requires a volunteer to pull out two numbers from a bag.

2. Ask the learners to write down the two numbers they think will be drawn.

3. The Lotto draw can be simulated by putting playing cards (in this case 1, 2, 3) in a bag and asking a volunteer to select two cards at random.

4. Draw the table below on the board and record the number of ‘winners’ and ‘losers’ in the first row.

“What percentage of the class won?”

“Is this what you expected? Why or why not?”

5. Encourage the learners to analyse the theoretical probability of winning. In 3-number Lotto, there are three combinations (1 and 2, 1 and 3, 2 and 3) since the order of the draw does not matter. Therefore the probability of winning is 1 out of 3 or 1/3 or 33.33%.

6. Discuss with the learners the methods that can be used to work out the number of possible outcomes. These include:

- using a tree diagram
- making an organised list:
- 2/3 x 1/2 = 2/6 or 1/3 (Note: There is a 2/3 chance in relation to the draw for the first number and a 1/2 for the second.)

7. Reflect back on the percentage of winners in the class simulation, ensuring the learners understand the theoretical probability is only a way of predicting what *may* occur, and it does not tell you what *will *occur.

8. Repeat steps 1–7 above with other versions of Lotto:

The key points to observe with the learners are that:

- the more numbers there are to choose from, the less the probability of a win
- the number of winners in the class (the experimental probability) is not usually the same as the theoretical probability.

9. When the learners understand how to calculate the probability of winning with the simple Lotto games, investigate the chance of winning firstdivision in Lotto (40 numbers, select 6).

“Which method can we use to find the number of possible outcomes?” (Hopefully the learners will realise that creating an organised list or using a tree diagram is not practical in this instance.)

Work together as a class to calculate the probability of winning first division:

6/40 chance of having the first number drawn

5/39 chance of having the second number drawn

4/38 chance of having the third number drawn

3/37 chance of having the fourth number drawn

2/36 chance of having the fifth number drawn

1/35 chance of having the sixth number drawn.

Thus there is a 6/40 x 5/39 x 4/38 x 3/37 x 2/36 x 1/35 = 720/2,763,633,600 = 1/3,838,380. (Note: this needs to be calculated on a computer as the number is larger than that permitted by most calculator displays.)

10. Discuss with the learners what a chance of 1 in 3.8 million means.

“What does a 1 in 3.8 million chance mean?”

“If you bought 10 lines a week would you be certain of a win in 380,000 weeks (and how many years is that)?”

“Would you make money if you bought enough lines to cover all the numbers?”

(no, it would cost 3,838,380 x 60 cents = $2,303,208)

## Follow-up activity

Investigate the chance of winning a first-division Lotto strike, a first division Lotto Powerball, or Big Wednesday.

Note: The New Zealand Lotteries Commission website www.nzlotteries.co.nz has information on the different Lotto games, including printable documents.

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