Transcript Document

using Tinkerplots
Ruth Kaniuk
Endeavour Teacher Fellow, 2013
1.13
AS 91038
Investigate a situation involving
elements of chance.
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…
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
distributions
see experimental probability
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.
Theoretical model
P(win) =
Toss1
H
Toss2
H
Toss3
H
Toss4
H
H
T
H
lose
H
H
T
T
win
H
T
H
H
lose
H
T
H
T
win
H
T
T
H
win
H
T
T
T
lose
T
T
T
T
T
H
T
lose
T
T
H
H
win
T
H
H
T
win
T
H
H
H
lose
T
H
T
H
win
T
H
T
T
lose
lose
lose
probability
1
8
1
16
1
16
1
16
1
16
1
16
1
16
1
8
1
16
1
16
1
16
1
16
1
16
1
16
6
16
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.
History of Results of Sampler 1
Options
count
10
5
0
-40
-20
0
20
40
60
profit
Circle Icon
Distribution of the profit from 100 games
(based on 100 simulations)
80
100
Distribution of the profit from 100 games
History of Results
Sampler
1
(basedofon
300 simulations)
count
30 0%
36%
Options
64%
20
10
0
-45
-35
-25
-15
-5
5
11.9167
15
25
profit
35
45
55
65
Circle Icon
36% of the time when 100 games are played, the profit
is zero or less.
Average profit per 100 games =$11.92
75
85
Use the model to investigate ‘what if’:
Investigate the likelihood of Bex’s class making a profit
different number of people play
if a
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
grids
[square eg 4x4 or rectangular grids]
different size
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 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
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.