PROBABILITY in the new curriculum AMA Statistics Day 1 September 2007 Louise Addison l.addison@auckland.ac.nz.

PROBABILITY in the new curriculum

AMA Statistics Day1 September 2007

Louise Addisonl.addison@auckland.ac.nz

http://events.stanford.edu/events/86/8619/

A thought provoker

What topic is this from? Where have I seen it before? What does it mean? What would a question look like? Is it Achieve / Merit / Excellence level? Could I draw a picture of it? What connections can I make? Do I need to find out more?

A Thought Provoker

What do you think…

P(X) = 0

Probability of Event X is 0

Pig Flying… Event X is impossible P(X’) = 1 If X = A B then A

and B are mutually exclusive

0 is the probability of Lotto rounded to 6 d.p.

Algebra?!

A couple of ads…

Link

NCEA Level One

AlgebraGeometryGraphs

NumberProbabilityTrigonometry

Put these topics in order from best done to worst done in Level One in 2006:

NCEA Level 1 2006

Not Achieved

Achieved Merit Excellence

Number 16 50 24 10

Trigonometry 26 47 22 5

Geometry 32 48 16 4

Algebra 37 34 18 10

Graphs 38 46 14 1

Probability 45 31 20 4

Why?

Sort the probability misconceptions cards into groups…

TRUE OR FALSE?

I’ve spun an unbiased coin 3 times and got 3 tails. It is more likely to be heads than tails if I spin it again.

TRUE OR FALSE?

Waikato plays netball against Auckland and can win, draw or lose. Therefore the probability Auckland will win is 1/3.

TRUE OR FALSE?

There are 3 red beads and 5 blue beads in a box. I pick a bead at random. The probability that it is red is 3/5.

TRUE OR FALSE?

I roll two dice and add the results. The probability of getting a total of 6 is 1/12 because there are 12 different possibilities and 6 is one of them.

TRUE OR FALSE?

There are more black balls in box A than in box B. If you choose 1 ball from each box you are more likely to choose a black ball from A than from B.

A B

TRUE OR FALSE?

Tomorrow it will either rain or not rain, so the probability that it will rain is 0.5.

TRUE OR FALSE?

Mr Brown is to have an operation. 90% of the people who have this operation make a complete recovery. There is a 90% chance that Mr Brown will make a complete recovery if he has this operation.

TRUE OR FALSE?

If six fair dice are thrown at the same time, I am less likely to get 1,1,1,1,1,1 than 1,2,3,4,5,6.

TRUE OR FALSE?

It is harder to throw a six than a three with a die.

TRUE OR FALSE?

Each spinner has two sections, one black and one white. The probability of getting black is 50% for each spinner

TRUE OR FALSE?

I flip two coins. The probability of getting heads and tails is 1/3 because I can get Head and Heads, Heads and Tails or Tails and Tails.

TRUE OR FALSE?

John buys 2 raffle tickets. If he chooses two tickets from different places in the book he is more likely to win than if he chooses the first two tickets.

TRUE OR FALSE?

13 is an unlucky number so you are less likely to win a raffle with ticket number 13 than with a different number.

13

TRUE OR FALSE?

It is not worth buying a lotto ticket with 1, 2, 3, 4, 5, 6 on it as this is less likely to occur than other combinations.

TRUE OR FALSE?

My granddad smoked 20 cigarettes a day for 60 years and lived to be 90, so smoking can’t be bad for you.

TRUE OR FALSE?

I have thrown an unbiased dice 12 times and not yet got a 6. The probability of getting a six on my next throw is more than 1/6.

Misconceptions about probability

All events are equally likely. Some events are less / more

likely than others (Representative Bias - 123456)

Later events may be affected by or compensate for earlier ones.(Recency Bias - BBBBBG)

When determining probability from statistical data, sample size is irrelevant.

Results of games of skill are unaffected by the nature of the participants.

Lucky/Unlucky numbers, etc. can influence random events.

In random events involving selection, results are dependent on numbers rather than ratios.

If events are random then the results of a series of independent events are equally likely, e.g. Heads Heads (HH) is as likely as Heads Tails (HT).

When considering spinners, probability is determined by number of sections rather than size of angles.

Curriculum key ideas

Investigating situations involving chance (L1-5) Exploring possible outcomes (L1-3) Acknowledging uncertainty (L2-3) Comparing experimental results with

expectations (L3-4), with theoretical results (L5) Acknowledging variation (L3-5) Acknowledging independence (L4-5) Using fractions and percentages (L4-5) and

ratios (L5) Comparing population parameters with those of

sample (L6)

Meteor Madness

Investigating situations involving chance (L1 - 5)

Exploring possible outcomes (L1-3) Acknowledging uncertainty (L2 - 3) Comparing experimental results with

expectations (L3-4), with theoretical results (L5).

Acknowledging variation (L3 - 5) Acknowledging independence (L4 - 5) Using fractions and percentages (L4-5)

and ratios (L5) Comparing population parameters with

those of sample (L6)

Rock / Paper / Scissors

Investigating situations involving chance (L1 - 5)

Exploring possible outcomes (L1-3) Acknowledging uncertainty (L2 - 3) Comparing experimental results with

expectations (L3-4), with theoretical results (L5).

Acknowledging variation (L3 - 5) Acknowledging independence (L4 - 5) Using fractions and percentages (L4-5)

and ratios (L5) Comparing population parameters with

those of sample (L6)

The truth of the matter

Investigating situations involving chance (L1 - 5)

Exploring possible outcomes (L1-3) Acknowledging uncertainty (L2 - 3) Comparing experimental results with

expectations (L3-4), with theoretical results (L5).

Acknowledging variation (L3 - 5) Acknowledging independence (L4 - 5) Using fractions and percentages (L4-5)

and ratios (L5) Comparing population parameters with

those of sample (L6)

Two way tables

Power point teaching tool

Rocket launch simulation

Spreadsheet

A couple of contexts

Rock Paper Scissors

Murphy’s Law

Correlation / causation

Power point

Discuss…

context

I graduated from Douglass College without distinction. I was in the top 98% of my class and damn glad to be there. I slept in the library and daydreamed my way through history lecture. I failed math twice, never fully grasping probability theory. I mean, first off, who cares if you pick a black ball or a white ball out of the bag? And second, if you're bent over about the color, don't leave it to chance. Look in the damn bag and pick the color you want. Plum, Stephanie

Hard Eight

Theoretical vs experimental probability

"In theory, there is no difference between theory and practice. But, in practice, there is." - Jan L.A. van de Snepscheut

A concluding thought…

Always be a little improbable. Oscar Wilde