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
probabilistic models.
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?
Randomness
What is randomness?
•
a lack of pattern or predictability in events
What does randomness look like?
file://localhost/Users/marionsteel/Desktop/workshops
Is this random scatter?
/probability workshop/random scatter.xls
Or is this random scatter?
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
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 deemphasize 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?
House Sales
What proportion of the houses that sold for over $600,000 were
on the market for less than 30 days?
Days on the market
Selling price
Less than
30 days
30 - 90 days
More than
90 days
Total
Under $300,000
39
31
15
85
$300,000 - 600,000
35
45
4
84
8
4
0
12
82
80
19
181
Over $600,000
Total
House Sales
What proportion of the houses that sold for over $600,000 were
on the market for less than 30 days?
Days on the market
Selling price
Less than
30 days
30 - 90 days
More than
90 days
Total
Under $300,000
39
31
15
85
$300,000 - 600,000
35
45
4
84
8
4
0
12
82
80
19
181
Over $600,000
Total
House Sales
What proportion of the houses that sold for over $600,000 were
on the market for less than 30 days?
Days on the market
Selling price
Less than
30 days
30 - 90 days
More than
90 days
Total
Under $300,000
39
31
15
85
$300,000 - 600,000
35
45
4
84
8
4
0
12
82
80
19
181
Over $600,000
Total
House Sales
What is the probability a house sold for under $300,000 given that
it sold in less than 30 days?
Days on the market
Selling price
Less than
30 days
30 - 90 days
More than
90 days
Total
Under $300,000
39
31
15
85
$300,000 - 600,000
35
45
4
84
8
4
0
12
82
80
19
181
Over $600,000
Total
House Sales
What is the probability a house sold for under $300,000 given
that it sold in less than 30 days?
Days on the market
Selling price
Less than
30 days
30 - 90 days
More than
90 days
Total
Under $300,000
39
31
15
85
$300,000 - 600,000
35
45
4
84
8
4
0
12
82
80
19
181
Over $600,000
Total
House Sales
What is the probability a house sold for under $300,000 given
that it sold in less than 30 days?
Days on the market
Selling price
Less than
30 days
30 - 90 days
More than
90 days
Total
Under $300,000
39
31
15
85
$300,000 - 600,000
35
45
4
84
8
4
0
12
82
80
19
181
Over $600,000
Total
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