J UNIOR P ROBABILITY

I NTRODUCTION  Grant Ritchie and Dr Michelle Dalrymple Cashmere High School  Junior Statistics focus  Collated information from variety of resources  Plan for this session…

T HINK 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 Steve is very shy and withdrawn, invariably helpful but with little interest in people or in the world of reality.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?

P P PDAC & P ROBABILITY Predict

C AT & M OUSE  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.

I NVESTIGATION …  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?

E XPECTATION  We can figure out who is likely to win without having to play a single game.  Imagine playing the game 32 times  1 st 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?

C HANGING THE G AME  Changing the dice allocation: which numbers make mouse go left/right.  Changing the game board.  Who is more likely to win now?

R EFLECTION …  Key teaching points?  Other directions the activity could lead?  Similar activities?

NZ C URRICULUM Junior Probability NCEA Subject content Values & Key Comps

NCEA – P ROBABILITY

C URRICULUM L EVEL T HREE L3 1/2 Stage L4 2 Stage L5 2/3 Stage + complex L6 complex + media Experimental Situations Theoretical Situations Realistic Situations no theory model model available  experimental dist n may vary from theory  Need to trial (sample) to get probability estimates  Different trials may give differing results  Outcomes are not always equally likely Key Ideas…  sample space represented by list, tables and tree diagrams.  students should be expected to predict the outcome and then reflect on this.  all outcomes  outcomes not always equally likely  derive probabilities and distributions.  often don’t know the exact probabilities

C URRICULUM L EVEL F OUR L3 1/2 Stage L4 2 Stage L5 2/3 Stage + complex L6 complex + media Experimental Situations Theoretical Situations Realistic Situations no theory model model available  Need to trial (sample) to get probability estimates  Different trials may give differing results  Outcomes are not always equally likely Key Ideas…  sample space represented by list, tables networks and tree diagrams.  students should be expected to predict the outcome and then reflect on this.  find P(event) =  estimates should be based on lots of trials  certain and impossible events  independence (informally)  all outcomes  outcomes not always equally likely  derive probabilities and distributions.  usually situations where probs are not known  use historical data to establish probability i.e. July 2009 temperatures  experimental dist n may vary from theory  compare experimental results with theoretical prob ies and dist ns

C URRICULUM L EVEL F IVE L3 1/2 Stage L4 2 Stage L5 2/3 Stage + complex L6 complex + media Experimental Situations Theoretical Situations Realistic Situations no theory model model available recognise the difference between these Key Ideas…  “is the game fair?”  variation is normal (no pun intended)  simple conditional situations  probabilities are between 0-1  calculate P(event) (rather than find)  still use large trials, but gain understanding of larger trial numbers yielding more representative results  certain and impossible events  independence (informally)  derive probabilities and distributions.  true probability  usually situations where probs are not known  use historical data to establish probability i.e. July 2009 temperatures  surveys  compare experimental results with theoretical prob ies and dist ns

E XPERIMENTAL /S IMULATIONS Level 3Level 4Level 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 L3 1/2 Stage L4 2 Stage L5 2/3 Stage + complex

E XPRESSING P ROBABILITIES Level 3Level 4Level 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 L3 1/2 Stage L4 2 Stage L5 2/3 Stage + complex

S AMPLE S PACE & O UTCOMES Level 3Level 4Level 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 E MPHASIS Year 9  Introducing PPDAC – through Probability  Statistical Literacy Year 10  Theoretical Probability & Literacy  PPDAC Statistics (Making the call)

P P PDAC & P ROBABILITY Predict

P ROBABILITY 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

P RISONERS 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

P RISONERS G AME Predict  What is the problem we are investigating?  What predictions do we want to make?

P RISONERS G AME  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….

P RISONERS G AME  Could we use probability theory to investigate this game?  How would you do this in your class?

P ROBABILITY DISTRIBUTION  How did we illustrate the probability distribution for our Prisoners game?

R EFLECTION …  Key teaching points?  Directions the activity could lead?  Similar activities?

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 S EA HORSE GAME

R ANDOMNESS  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?

R ANDOMNESS

R ANDOMNESS R EFLECTION …  Level 6  What should we be exposing our students to before this?  ch?v=YpvE0Co66nU ch?v=YpvE0Co66nU Start – 1.20

T HINK 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?

S KUNK  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.

S KUNK  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

S KUNK

R EFLECTION …  Key teaching points?  Directions the activity could lead?  Similar activities?

K EY 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

A NY QUESTIONS …

W ORKSHOP FEEDBACK … On a piece of paper please give us some feedback It 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?....