JUNIOR PROBABILITY

INTRODUCTION

Grant Ritchie and Dr Michelle Dalrymple Cashmere High School

Junior Statistics focus

Collated information from variety of resources

Plan for this session…

THINK ABOUT….

1. Steve is very shy and withdrawn, invariably helpful but with little interest in people or in the world of reality. A meek and tidy soul, he has a need for order, structure and a passion for detail. Is it more likely that Steve is a librarian/ sales assistant/ primary teacher?

2. Which is more likely when I throw a coin 6 times & record the results: H-T-H-T-T-H or H-H-H-T-T-T

3. In a casino a roulette wheel has stopped on red in the last 7 games. Would you rather bet on red or black in the next game?

PPPDAC & PROBABILITY

Predict

CAT & MOUSE

Demonstration of the game You are the mouse. Each roll of the dice you

move one space according to whether the roll is Odd (O) or Even (E). Supposing the 'cat wins'. Does this mean the

cat will win every time?

How can we find out which has the better chance - the Cat or the Mouse? The obvious answer is to play many games.

http://www.youtube.com/watch?v=VG02xeBz8PE

http://www.youtube.com/watch?v=Ck6Iys1l2qw

INVESTIGATION…

Question: Is the mouse more likely to get eaten or get the cheese?

Predict: What do you think will happen? Plan: Play 10 Games and record your results Data: Count up your results & add them to the class data. Analysis: What do you notice?

What was the overall probability that the mouse/ cat won?

How come there were anomalies Is this game fair??? How can we tell?

(software simulation of MANY games!!) Conclusion: answer your question, how could we extend

our investigation?

EXPECTATION

We can figure out who is likely to win without having to play a single game.

Imagine playing the game 32 times

1st step has two possible roads: in how many of the 32 games would you expect mouse to go down either road?

Now of these games, how many times would you expect mouse to go down each road?

CHANGING THE GAME

Changing the dice allocation: which numbers make mouse go left/right.

Changing the game board.

Who is more likely to win now?

REFLECTION…

Key teaching points?

Other directions the activity could lead?

Similar activities?

NZ CURRICULUM

Junior Probability

NCEA

Subject content

Values & Key Comps

NCEA – PROBABILITY

Junior Probability

Chance & Data

Probability

Probability Distributions

Stats Reports

Stats Reports

Experimental Probability

Experiments

Experiments

Simulations

EXPERIMENTAL/SIMULATIONS

Level 3 Level 4 Level 5

• Carry out simple experiments (e.g flip a coin)- notice that results vary between trials

• Accept variation between samples and expectations

S3-3

• Compare expected distributions and experimental distributions (e.g roll 2 dice)

• Graph results to see distributions and compare them to the theoretical models.

S4-3

• Compare expected distributions and trial distributions (e.g roll a die and flip a coin)

• Graph results to see distributions and compare them to the theoretical models.

S5-3

L31/2 Stage

L42 Stage

L52/3 Stage + complex

EXPRESSING PROBABILITIES

Level 3 Level 4 Level 5

• Written in words e.g. likely certain etc

S3-3

• Simple Fractions, percentages and decimals e.g. using number lines between 0-1 to show probabilities.

S4-4

• Express as fraction percentages and decimals

• Calculate probability using equally likely outcomes

S5-4

L31/2 Stage

L42 Stage

L52/3 Stage + complex

SAMPLE SPACE & OUTCOMES

Level 3 Level 4 Level 5

• In simple situations use

- lists- basic tree

diagrams,- tables

to show possible outcomes

S3-3

• Two-stage situations (eg possible outcomes in a 2 child family) use

- Lists,- Tree

diagrams,- Networks,- 2 way tables

to show possible outcomes

S4-3

• Two-stage situations with different events(eg possible outcomes main meal and dessert selections) use

- Lists,- Tree

diagrams,- Networks,- 2 way tables

to show possible outcomes

S5-4

CHS EMPHASIS

Year 9

Introducing PPDAC – through Probability

Statistical Literacy

Year 10

Theoretical Probability & Literacy

PPDAC Statistics

(Making the call)

PPPDAC & PROBABILITY

Predict

PROBABILITY DISTRIBUTIONS

The probabilities associated with each outcome DISCRETE or CATEGORICAL outcomes

Table, Graph (eg dot plot, bar chart)

MEASUREMENT data Table (with class intervals) Graph (eg histogram)

At higher curriculum levels we can also use formulas

PRISONERS GAME

1. Place 10 prisoners (x) in any cells.

2. If their cell number is rolled they escape: cross them out.

3. When all escaped you (the big boss) wins.

0 1 2 3 4 5

PRISONERS GAME

Predict

What is the problem we are investigating?

What predictions do we want to make?

PRISONERS GAME

Roll your dice and record the difference Repeat 60 times (work together at your table…) Add your data to the class’

What do you notice? What do you wonder?

1. If you were a prisoner, which cell would you want to be in and why?

2. How could we find the probability of different totals? Computer simulation….

PRISONERS GAME

Could we use probability theory to investigate this game?

How would you do this in your class?

PROBABILITY DISTRIBUTION

How did we illustrate the probability distribution for our Prisoners game?

REFLECTION…

Key teaching points?

Directions the activity could lead?

Similar activities?

1

2

0 3

4

5

FINISH

There are 6 seahorses racing, one in each lane

Which seahorse has the best chance of winning?

Take turns to roll two dice and work out the difference between the two dice.

The seahorse with that dice difference moves one place forward

Repeat Record which horse wins the game

SEA HORSE GAME

RANDOMNESS

On your piece of paper in front of you…

Without looking at what anyone else is doing…

Randomly place one ‘x’ on it

Now

Leave your piece of paper on your desk

Have a look at everyone else’s ‘x’

What do you notice?

RANDOMNESS

RANDOMNESS REFLECTION…

Level 6

What should we be exposing our students to before this?

http://www.youtube.com/watch?v=YpvE0Co66nU

Start – 1.20

THINK ABOUT…

I might make more money if I was in business for myself; should I quit my job?

An fire might destroy my house; should I buy insurance?

My mathematics teacher might collect homework today; should I do it?

SKUNK

Each letter of SKUNK represents a different round of the game; play begins with the "S" column and continue through the "K" column. The object of SKUNK is to accumulate the greatest possible point total over five rounds. The rules for play are the same for each of the five rounds.

At the beginning of each round, every player stands. Then, a pair of dice is rolled. (Everyone playing uses that roll of the dice; unlike other games, players do not roll the dice for just themselves.)

A player gets the total of the dice and records it in his or her column, unless a "one" comes up. If a "one" comes up, play is over for that round and all the player's points in that column are

wiped out. If "double ones" come up, all points accumulated in prior columns are wiped out as well. If a "one" doesn't occur, the player may choose either to try for more points on the next roll (by

continuing to stand) or to stop and keep what he or she has accumulated (by sitting down). Note: If a "one" or "double ones" occur on the very first roll of a round, then that round is over

and each player must take the consequences.

SKUNK

1 = Round over, points for round wiped out if still standing

1,1 = Round over, ALL points from prior rounds wiped out if still standing

SKUNK

REFLECTION…

Key teaching points?

Directions the activity could lead?

Similar activities?

KEY IDEAS…

Appreciation of uncertainty/randomness

Lots and lots and lots of probability ideas out there Make sure you know what your

key teaching points are… Make sure you’re aware of the

directions you can run with them…

Remember to include situations where comparing with theory isn’t possible

Remember we still need to teach mathematical probability as well as from the experimental perspective

ANY QUESTIONS…

WORKSHOP FEEDBACK…

On a piece of paper please give us some feedbackIt can be for about these workshops, or for CMA generally

1. Please keep doing…

2. Please stop doing…

3. Please start doing…

4. Anything else you want us to know?....