1. using Tinkerplots Ruth Kaniuk Endeavour Teacher Fellow, 2013

2. Investigate a situation involving elements of chance. 1.13 AS 91038

3. Why use Tinkerplots? …AS91038 Compare and describe the variation between theoretical and experimental distributions in situations that involve elements of chance. Investigate situations that involve elements of chance: comparing discrete theoretical distributions and experimental distributions, To create sufficient data before the boredom of throwing dice sets in…

4. AS 91038.. gathering data by performing the experiment selecting and using appropriate displays including experimental probability distributions comparing discrete theoretical distributions and experimental distributions as appropriate To be able to see experimental probability distributions

5. appreciating the role of sample size.. How do they appreciate sample size if they only get to n = 50 before…

6. To appreciate that a probability has a fixed value, but that the chance event it is describing is not so certain.

7. ‘the expected’ does not always occur

8. To develop a better appreciation for uncertainty

9. Fundraiser 1 (Tinkerplots) Adapted from http://nrich.maths.org/848http://nrich.maths.org/848 The school is having a fundraising fair. Each class is responsible for organising one activity. Bex suggests that her class run a game as follows…

10. THE GAME: Start with a counter on the star in the grid below. Toss a coin. Move up for Head, left for Tail. Keep tossing the coin until you are either off the board (lose) or you have won by reaching the top left square on the grid.

11. I wonder: If it costs $2 to play and the player gets $5 if she/he wins (a gain of 5-2 = $3), will Bex’s class make a profit, assuming 100 people each play the game once?

12. Please play the game Play 5 times each. Tables combine your results, then estimate the profit if 100 games were played.

13. Out of 16 games, the player is expected to win 6 times and lose 10 times. Bex’s class would get 10 x 2 and pay out 6 x 3 Bex’s class would make $2 for every 16 games. If 100 people played then Bex’s class would make about 6 x 2 =$12 profit. P(win) = Theoretical model

14. Distribution of the profit from 100 games (based on 100 simulations)

15. 36% of the time when 100 games are played, the profit is zero or less. Average profit per 100 games =$11.92 Distribution of the profit from 100 games (based on 300 simulations)

16. Use the model to investigate ‘what if’: Investigate the likelihood of Bex’s class making a profit if a different number of people play the game (what happens if fewer people play, what happens if more people play) OR Investigate a suitable prize and cost to play that so that the risk of Bex’s class losing money is reduced (and the cost to play is small enough that people are likely to play) OR Investigate possible profit from the game using different size grids [square eg 4x4 or rectangular grids]

17. So… why use simulation To get an idea of what ‘long run’ means In the long run we would expect a profit of about $12 from 100 people playing… But understand that there is uncertainty around that expected value The expected value has a distribution around it If 100 people played the game I could lose money (maybe $45) or I could make money (maybe $80) but I am more likely to make between…

18. So… why use simulation… To use probability models to mimic the real world To use the model to ask ‘what if?’ – what are the likely impacts of a change To introduce students to how applied probabilists think and work

19. This work is supported by: The New Zealand Science, Mathematics and Technology Teacher Fellowship Scheme which is funded by the New Zealand Government and administered by the Royal Society of New Zealand and Department of Statistics The University of Auckland

20. Challenge! Your task is to design a game which will make a profit. Your game may use dice, coins or a spinner. How much does it cost to play? How is the game played? What is the prize ( or prizes)? What is the probability of winning? How much money do you expect to make if 100 people play?

21. Which is the better bet? You pay $1 to play each game. Game 1: 4 coins are tossed. You win $3 if the result is two heads and two tails. Game 2: 3 dice are rolled. You win $2 for each 6 that appears.