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Chapter 5: Probability
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? |
Randomness and
Probability Models
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State the basis for all predictions based on probability models. Relevance: The law of large numbers is arguable
the single most important concept in probability and statistics.
The Law of Large Numbers--as the number of data
points or observations increase, the actual results will tend to
converge on the results predicted by probability models.
Probability model
predictions tend to be reliable only for large groups.
example:
Probability models predict
that the thousands of people gambling each day in Las Vegas
will, on average, lose money. Probability model predictions for
an individual are much less reliable. Some individuals do indeed
win.
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Name the two factors which exist in a
random phenomenon.
- Uncertain outcome
- Regular distribution of outcomes
with a large number of trials. (Note that even randomness follows
a pattern
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State the
differences between
outcomes,
events,
and sample spaces.
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Use tree
diagrams to identify sample spaces.
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Use the
multiplication rule to calculate the number of outcomes.
- Number
of ways to do task1 = a
- Number
of ways to do task2 = b
- Number
of ways to do task1 and task2 = a x b
Homefun
(formative/summative assessment): Read section 5.1
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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? |
Probability Models
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Calculate
the number of outcomes using sampling without replacement.
(Hint: use a tree diagram.)
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Draw a
Venn diagram for
disjointed events and give examples.
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Draw a
Venn diagram for
independent events
and give examples. For independent events, knowing that one has
occurred does not yield any additional information about whether the
other event has occurred.
example:
Knowing that Jane is a girl yields no additional information about
whether she does or does not have brown eyes. Hence, the probability
of being a girl is independent of the probability of having brown
eyes.
Note:
disjointed events are
not
independent. Knowing that one of a pair of disjointed events has
occurred makes it certain the other has not.
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Correctly
apply the 5 probability rules
- 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)
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Draw a Venn diagram showing
the complement
(Ac) of an event (A). (note the complement of A = not A)
Homefun
(formative/summative assessment): Read
section 6.2
Relevance: Converting a
problem into a picture such as a Venn diagram engages your right brain in
problem solving. It considerably boosts your problem solving skill. |
- 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 you
marry Mr./Ms. right (Three Coins in the Fountain)? |
Creating Probability Models
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Calculate the
probabilities of events for equally likely outcomes.
- drawing an
integer ( 0-9 ) from a hat
- flipping coins
- rolling a die
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P(A) =
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count of outcome
in A |
| count of outcomes in
S |
- Draw probability distributions for coin flipping and similar
events. Note that the probability
distribution is a picture of the probability model. It can be used
for answering what-if questions.
- Describe how
mathematical models of dice, coins, or drawing
numbers from a hat can be used for making predictions about real
world events. Making predictions is
essentially the same thing as answering what-if questions.
Relevance: The gambling
industry in Las Vegas was built on probability models, all of which depend
on the Law of Large Numbers.
formative assessment: draw various distributions on the
white-boards (teams of 4) |
Lesson 3
Key Concept:
Ss.
Purpose: Rs.
Warm up (Individual):
c5.
Interactive Discussion:
Problem Solving (Teams of two):
cd.
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| Essential Question:
Can you simulate the probability
of marrying Mr. / Ms. right? |
Simulations
vs. Probability Models
- Compare the similarities and differences
between a prediction based on a simulation and one based on a
mathematical model.
Mathematical model: built from equations and used for
making predictions about real life events without using a random
number generator. Mathematical models always
give the same answer.
Simulation:
makes predictions about real life events by repetitively using a
random number generator.
Results depend on making use of the
Law of Large Numbers.
Simulations usually give slightly different results
when they are repeated.
- Describe a Monte Carlo simulation.
These are almost like running an actual
experiment.
a
computerized simulation - like all simulations, uses random
number generators.
uses algorithms - sets of
computer instructions (lines of code) used for problem solving.
Algorithms tend to be more flexible than
equations.
- Perform simulations.
Relevance: Monte Carlo
simulations have replaced all forms of nuclear bomb testing. They are
useful for a wide range of simulations from cell phone use to predictions
about pandemics.
formative assessment: use 3 coins to simulate a distribution.
Show results on the white-boards (teams of 4) |
Lesson 3
Key Concept:
Ss.
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:
What is the probability
that you can make a correct decision given partial information and
what are the ramifications for group decisions and democracy? |
Using Probability Models to
Understand Real World Events
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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).
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Model the expected decision
making accuracy for:
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unanimous
decisions
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majority
rules--voting
Relevance: The problem-solving
power of any form of mathematics comes from its ability to model real
world applications. It's amazing that probability can accurately model
small group dynamics. Psychological studies have tended to verify the
conclusions drawn from the probability equations. The psychologists, of
course, used statistics to analyze the their data.
Homefun
(formative/summative assessment): 6.23, 6.25
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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
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Draw the Venn
diagram and define a union
(A or B) or intersection (A
and B) for a collection of events.
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Calculate the
probabilities for unions (unions correspond to
or-statements) with independent and disjointed events.
Homefun
(formative/summative assessment): 6.27, 6.31
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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
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
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Calculate
conditional probabilities for unions ( contain
or-statements,
p.348-349).
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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
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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)
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Use trees for
probability calculations with dependent events. P(B|A)
≠ P(B)
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Use Venn Diagrams and
probability equations to explain the differences between the
following types of events. Note:
disjointed ≠ independent ≠ conditional.
In other words the 3 different categories are completely separate.
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disjointed
-- events are mutually exclusive.
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independent
-- the occurrence of one event gives no additional information about
the other.
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conditional
-- the occurrence of one event gives additional information about the
other.
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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
(formative/summative assessment):
1, 3, 11, 15, 23 pp. 293-296
45, 47, 57, 59 pp. 309-311
63, 71, 87, 97, 105 pp. 329-333
Relevance: Venn diagrams help
us visualize probability problems but tree diagrams are and an even more
powerful way to not just visualize but solve probability problems.
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Essential Question:
How can I make an
"A" on the test? |
Probability Review
- Master the vocabulary
- Correctly draw
tree diagrams and relate them to
Venn diagrams.
- Work the practice test.
- Review the objectives.
Homefun (formative/summative assessment): Chapter 5 AP Statistics Practice Test, do multiple choice and free response
Summative Assessment:
Unit Exam objectives 1- 26
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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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