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### Probability level 8 NZC

AS91585 Apply probability concepts in solving problems

NZC level 8

Investigate situations that involve elements of chance

- calculating probabilities of independent, combined, and conditional events

AS 3.13 Apply probability concepts in solving problems

WHAT ISN’T INCLUDED?

AS 3.13 Apply probability concepts in solving problems

WHAT ISN’T INCLUDED?

No combinations, no permutations,

No formal questions on expected value. This means that students are expected to bring an understanding of expected value from NCEA level 1 and 2. They may need to calculate a simple expected value as part of solving a problem, but will not be asked questions such as “calculate the expected value”.

AS 3.13 Apply probability concepts in solving problems

WHAT IS INCLUDED?

Methods include a selection from those related to:

- true probability versus model estimates versus experimental estimates
- randomness
- independence
- mutually exclusive events
- conditional probabilities
- probability distribution tables and graphs
- two way tables
- probability trees
- Venn diagrams.

AO S8-4 TKI

A. Calculating probabilities of independent, combined, and conditional events:

- Students learn that some situations involving chance produce discrete numerical variables, that situations involving real data from statistical investigations can be investigated from a probabilistic perspective. These have probability distributions. They can be investigated by making assumptions about the situation and applying probability rules and/or by doing repeated trials of the situation and collecting frequencies.

AO S8-4 on TKI

- Selects and uses appropriate methods to investigate probability situations including experiments, simulations, and theoretical probability, distinguishing between deterministic and probabilisticmodels.
- Interprets results of probability investigations, demonstrating understanding of the relationship between true probability (unknown and unique to the situation), model estimates (theoretical probability), and experimental estimates.
- Selects and uses appropriate tools to solve problems in probability, including two-way tables, Venn diagrams, and tree diagrams, including combined events.
- Solves probability problems involving conditional probabilities, randomness, independence, and mutually exclusive events.

Randomness

Students are expected to be familiar with the behaviour of random variables and the appearance of random distributions.

What is randomness?

What does randomness look like?

How can we teach it?

How can we assess it?

What does randomness look like?

file://localhost/Users/marionsteel/Desktop/workshops/probability workshop/random scatter.xls

How can we teach an understanding of randomness?

- Lots of hands on experience with random variables
- Games like Fooling the teacher
- Encourage students to confront their own misconceptions and fallacies about probability and randomness
- Teach it from year 9 onwards so that students have developed a sound understanding of it by the time they reach year 13.

How can we assess “methods relating to” randomness?

- Methods relating to randomness are virtually all the methods of probability and statistics.
- Students might be asked to justify strategies or decisions, which might include reference to random outcomes or probabilities of random variables.

true probability versus model estimates versus experimental estimates

What is the probability that the next baby born in NZ will be a boy?

We start with a basic model based on our previous knowledge and experience. With more information, we can improve our model.

true probability versus model estimates versus experimental estimates

What is the probability that a biased coin will land heads up?

We start with a model (null hypothesis) of landing equally likely on heads and tails.

We look at data, asking the question whether it provides evidence that our model is not a good representation of the real world.

Deterministic and probabilistic models

- A deterministic model does not include elements of randomness. Every time you run the model with the same initial conditions you will get the same results.
- A probabilistic model does include elements of randomness. Every time you run the model, you are likely to get different results, even with the same initial conditions.

Waiting times

A simple model of a cash machine

- Customers arrive every two minutes, on average.
- Customers take 2 minutes to use the machine.
- What is the probability that a customer has to wait 3 minutes or more?

Waiting times

In a deterministic model people arrive every two minutes and use the machine. There is no waiting time.

We can use a simulation to investigate waiting times for a probabilistic model. We can simulate 15 random arrival times in a 30 minute period:

2 4 5 5 10 11 12 15 16 19 20 24 29 29 29

Modelling waiting times

From our simulation, 2/15 customers waited 3 minutes or more.

Our estimate of the probability that a customer waits 3 minutes or more is 0.13.

The following slides are from Auckland Statistics Day 2004 (apologies to the University of Auckland Statistics Department as their logo wouldn’t copy).

Since 2004, we have been encouraged to de-emphasize Venn diagrams for solving probability problems. Two way tables are a much more effective problem solving method, and should be student’s first choice.

What progress has been made towards that shift in teaching practice?

Less than

30 days

30 - 90 days

More than

90 days

Selling price

Total

39

31

15

85

Under $300,000

35

45

4

84

$300,000 - 600,000

8

4

0

12

Over $600,000

Total

82

80

19

181

House SalesWhat proportion of the houses that sold for over $600,000 were on the market for less than 30 days?

Less than

30 days

30 - 90 days

More than

90 days

Selling price

Total

39

31

15

85

Under $300,000

35

45

4

84

$300,000 - 600,000

8

4

0

12

Over $600,000

Total

82

80

19

181

House SalesWhat proportion of the houses that sold for over $600,000 were on the market for less than 30 days?

Less than

30 days

30 - 90 days

More than

90 days

Selling price

Total

39

31

15

85

Under $300,000

35

45

4

84

$300,000 - 600,000

8

4

0

12

Over $600,000

Total

82

80

19

181

House SalesWhat proportion of the houses that sold for over $600,000 were on the market for less than 30 days?

Less than

30 days

30 - 90 days

More than

90 days

Selling price

Total

39

31

15

85

Under $300,000

35

45

4

84

$300,000 - 600,000

8

4

0

12

Over $600,000

Total

82

80

19

181

House SalesWhat is the probability a house sold for under $300,000 given that it sold in less than 30 days?

Less than

30 days

30 - 90 days

More than

90 days

Selling price

Total

39

31

15

85

Under $300,000

35

45

4

84

$300,000 - 600,000

8

4

0

12

Over $600,000

Total

82

80

19

181

House SalesWhat is the probability a house sold for under $300,000 given

that it sold in less than 30 days?

Less than

30 days

30 - 90 days

More than

90 days

Selling price

Total

39

31

15

85

Under $300,000

35

45

4

84

$300,000 - 600,000

8

4

0

12

Over $600,000

Total

82

80

19

181

House SalesWhat is the probability a house sold for under $300,000 given

that it sold in less than 30 days?

Solving probability problems

- Encourage the use of a two way table as the first method to consider.
- Encourage flexibility. Solve the same problem using:
- Two way tables
- Tree
- Venn diagram
- Probability algebra

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