Probability

using Tinkerplots

Ruth Kaniuk

Endeavour Teacher Fellow, 2013

Investigate a situation involving

elements of chance.

1.13 AS 91038

Why use Tinkerplots?

…AS91038Compare 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…  

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

appreciating the role of sample size..

How do they appreciate sample size

if they only get to n = 50

before…

To appreciate that a probability has

a fixed value,

but that the chance event it is

describing is not so certain.

‘the expected’

does not always occur

To develop a better appreciation

for uncertainty

Fundraiser 1 (Tinkerplots)Adapted from http://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…

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.

Win!

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?

Please play the game

Play 5 times each.

Tables combine your results, then estimate the profit if 100 games were played.

Toss1 Toss2 Toss3 Toss4 probability H H H lose 1

8

H H T H lose 116

H H T T win 116

H T H H lose 116

H T H T win 116

H T T H win 116

H T T T lose 116

T T T lose 18

T T H T lose 116

T T H H win 116

T H H T win 116

T H H H lose 116

T H T H win 116

T H T T lose 116

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) = 616

Theoretical model

-40 -20 0 20 40 60 80 100

0

5

10

count

profit

Circle Icon

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

36% of the time when 100 games are played, the profit is zero or less. Average profit per 100 games =$11.92

-45 -35 -25 -15 -5 5 15 25 35 45 55 65 75 85

0

10

20

30 0% 36% 64%

11.9167

count

profit

Circle Icon

Distribution of the profit from 100 games(based on 300 simulations)

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]

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…

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

This work is supported by:

The New Zealand Science, Mathematics and Technology Teacher Fellowship Schemewhich is funded by the New ZealandGovernment and administered by the Royal Society of New Zealand

and      Department of StatisticsThe University of Auckland

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?

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.