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I II. Anticipating Patterns:
Exploring random phenomena using probability and simulation (20%
–30%)
A. Probability
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Interpreting probability,
including long-run relative frequency interpretation
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“Law of Large Numbers”
concept
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Addition rule,
multiplication rule, conditional probability, and independence
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Discrete random variables
and their probability distributions, including binomial and
geometric
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Simulation of random
behavior and probability distributions
B. Combining independent
random variables
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Notion of independence
versus dependence
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Mean and standard
deviation for sums and differences of independent random variables
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Objectives |
| Essential Question:
Can you win money in Las Vegas? |
Ch 6.1, 6.2 -- Randomness and
Probability Models
-
State the basis for all predictions based on probability models.
The Law of Large Numbers
-
Name the two factors which exist in a
random phenomenon (p. 314).
- Uncertain outcome
- Regular distribution of outcomes
with a large number of trials. (Note that even randomness follows
a pattern.)
- State the
difference between an
outcome, an
event (p. 324),
and a sample
space (p. 318).
-
Use tree
diagrams to identify sample spaces.
-
Use the
multiplication rule to calculate the number of outcomes.
Homefun: 6.8 --
Read section 6.1, 6.2
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Activities |
- Lesson 1
- Key Concept: Predictions
from probability models tend to match results from real
data if the sample size is large enough.
- Purpose: Learn the basic principles and vocabulary of
probability.
Interactive Discussion:
Problem solving
(individual):
- Draw the tree diagram of
outcomes for rolling a pair of dice.
- Draw a tree diagram for the sample space obtained by
throwing one coin and one die. Identify an event and calculate the
probability that it will occur. Are the various events independent?
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| Essential Question:
Is anything in nature
truly random? |
Ch 6.2 -- Probability Models
-
Calculate
the number of outcomes using sampling without replacement.
(Hint: use a tree diagram.)
-
Draw a
Venn diagram for
disjointed events and give examples.
-
Draw a
Venn diagram for
independent events
and give examples.
-
Correctly
apply the 5 probability rules (p. 324-325 and p.341)
- Range of values: 0 to 1
- Sum of probabilities for all outcomes =
1.
- Prob. of not happening = 1-(prob. of happening)
- Addition Rule -
2 disjointed events, prob. of one
or the other occurring is the sum of individual probabilities.
P(A or B) =
P(A) + P(B)
- Multiplication Rule-
2 independent events, the probability of one
and
the other occurring is the probabilities of both multiplied
P(A and B) =
P(A) P(B)
-
Draw a Venn diagram showing
the complement
(Ac) of an event (A).
Homefun: 6.8 6.9, 6.11, 6.13,
-- Read
section 6.3
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- Lesson 2
- Key Concept:
Probabilities for independent events can be calculated using a
few simple rules.
- Purpose:
Be able to Calculate Probabilities for independent events
Interactive Discussion:
10 monkeys are in a barrel, 3 are dead and 7 are
alive. What is the probability of removing a dead monkey? After 3
dead monkeys have been removed (and not replaced), what is the
probability of removing an additional dead monkey. Run the same mind
experiment with replacement.
Problem Solving (Teams of
two): Calculate the probability of hopeless failure for small
groups of 2-7. Assume that the odds of a single person being right
is 60%.
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Stats
Investigation: The Spinning Wheel
(p. 310 Teams of two) |
| Purpose:
Determine if actual results match predicted results better
with large sample sizes.
Instructions: Read the
instructions on p. 310. Use a TI-83 and a die to generate
groups of 3 random numbers. Record 20 groups of random numbers
for each type of randomization. Calculate the probability of
getting at least one number in the correct order for each set
of 20 experiments and record the probabilities on the board.
Calculate average and range for the two data sets from
individual teams. Note: the two averages are also calculated probabilities.
Use a tree diagram to list the
possible outcomes. From the tree diagram, calculate the
theoretical probability of having at least one of 3 numbers in
order for the above experiment.
Questions /Conclusions:
- Was there a significant
difference (in other words one probably not due to random
chance) in the calculated probabilities for the two different types of random
number generators?
- How closely did the
calculated probability of the entire group match the predicted?
- How did the calculated
probabilities from individual data
sets (n=20) compare to the three probabilities calculated
by combining data from all the groups?
Resources/Materials: TI-83's,
10 dice |
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| Essential Question:
What is the probability that you
can make a correct decision given partial information and what are the
ramifications for group decisions and democracy? |
Ch 6.2 -- Probability Models
-
Calculate the
probabilities of events for equally likely outcomes.
| P(A) =
|
count of outcome in
A |
| count of outcomes in S |
-
Create
distributions representing the sample space of a random process both
for outcomes with equal probability and outcomes with unequal
probabilities (see example 6.6, p. 320 and see
How to
Design Small Decision Making Groups).
-
Model the expected decision
making accuracy for:
-
unanimous
decisions
-
majority
rules--voting
Homefun: 6.31, 6.35
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Lesson 3
Key Concept:
Sample spaces can be represented using distributions.
Purpose: Relate
distributions to tree diagrams and sample spaces.
Warm up (Individual):
calculate the probabilities of getting 3 heads with 3 coin
tosses, four heads with 4 coin tosses and 5 with 5.
Interactive Discussion:
Problem Solving (Teams of two):
create 2 distributions for the decision accuracy of 4 person
groups assuming individual accuracies of 50% for the first
distribution and 60% for the second.
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| Essential Question:
Are all disjointed events
independent? |
Solving Probability
Problems for Unions and intersections given independent or disjointed
events
-
Define a union
(A or B) or intersection (A
and B) for a collection of events (p.341).
-
Calculate the
probabilities for unions (unions correspond to
or-statements,
p.341-343) with independent
and disjointed events.
Homefun: 6.37, 6.39
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- Lesson 4
- Key Concept:
Probabilities with Unions
- Purpose:
Solve probability problems with both disjointed and
non-disjointed independent events.
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Interactive Discussion:
Objective 16. Use coin and election (control of House, Senate)
examples |
| P(A or B) = P(A) + P(B) -
P(A and
B) |
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- Disjointed: 2 coins both
heads or both tails
- Not Disjointed: 2 coins
at least one head or at least one
head tail
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Problem Solving (Teams of
two): Work problems 6.39 and 6.40
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| Essential Question:
What are conditional
probabilities? |
Solving Conditional
Probability Problems for Unions and intersections
-
Calculate
conditional probabilities for unions ( contain
or-statements,
p.348-349).
-
Calculate
conditional
probabilities for intersections ( contain and-statements,
p.348-349).
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Determine if 2 events are independent
using the test P(B|A) = P(B).
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- Key Concept:
Conditional Probability for unions and intersections
- Purpose:
Solve probability problems when the events are not
independent. In other words they involve events which are
conditional.
| Interactive Discussion:
Objective 17. poor, not poor; go to college, not go to college. |
| P(A
and
B) = P(A) * P(B|A) |
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| P(B|A) = P(A
and B)/P(A)
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Used as test for
independence.
When P(B|A) = P(B) the events are independent. |
Problem Solving (Teams of
two): Work problems 6.41, 6.46, 6.48
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| Essential Question:
What are the dangers of massive
drug abuse testing programs ? |
- Solving Probability Problems With Diagrams
- Why
you
- should be a tree hugger
-
Use trees for
probability calculations (see
The
Probability of Penalizing the Innocent Due to Bad Test Results)
with independent events. P(B|A) = P(B)
-
Use trees for
probability calculations with dependent events. P(B|A)
≠ P(B)
-
Use Venn Diagrams and
probability equations to explain the differences between the
following types of events
-
disjointed
-
independent
-
conditional
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Relate Venn Diagrams to
tree diagrams. (the equations and trees give the same
information. Both can be related to Venn Diagrams)
Homefun:
6.41, 6.43, 645, 6.49,6.53, 6.55
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- Key Concept: Predictions
from probability models tend to match results from real
data if the sample size is large enough.
- Purpose: Learn the basic principles and vocabulary of
probability.
Interactive Discussion:
Board Work
(individual): Solve tree diagram problems
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