Mr. Rogers - AP Statistics Objectives
 Syllabus 1st Quarter 2nd Quarter 3rd Quarter 4th Quarter Probability Binomal Distr Sampling Distr Conf Intervals
 Unit Plan Practice Test
Latin
 Latin/Greek Root Words arch--------->ancient, example: archtype;         chrono------>time, example: chronology;             -dom----------->quantity/state, example: freedom               fer-------->carry, example: transfer;               gen--------->birth, example: generate;                 luc-------->light, example lucid;                 neo--------->new, example: neonatologist;                olig--------->few, example: oligarchy;              omni--------->all, omniscient;            sym--------->together, symbol; (Statistics connection)

Chapter 5: Probability

II. Anticipating Patterns: Exploring random phenomena using probability and simulation (20% –30%)

A. Probability

1. Interpreting probability, including long-run relative frequency interpretation

2. “Law of Large Numbers” concept

3. Addition rule, multiplication rule, conditional probability, and independence

4. Discrete random variables and their probability distributions, including binomial and geometric

5. Simulation of random behavior and probability distributions

B. Combining independent random variables

1. Notion of independence versus dependence

2. Mean and standard deviation for sums and differences of independent random variables

Objectives

 Essential Question: Can you win money in Las Vegas?

Randomness and Probability Models

1. 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.

1. 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
1. State the differences between outcomes, events and  sample spaces.

2. Use tree diagrams to identify sample spaces.

3. 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

 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?
 Essential Question: Is anything in nature truly random?

Probability Models

1. Calculate the number of outcomes using sampling without replacement. (Hint: use a tree diagram.)

2. Draw a Venn diagram for disjointed events and give examples.

3. 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.

1. 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)

1. 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%.

 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
 Essential Question: What is the probability you marry Mr./Ms. right (Three Coins in the Fountain)?

Creating Probability Models

1. Calculate the probabilities of events for equally likely outcomes.

• drawing an integer ( 0-9 ) from a hat
• flipping coins
• rolling a die
 P(A) = count of outcome in A count of outcomes in S
1. 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.
2. 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.

 Essential Question: Can you simulate the probability of marrying Mr. / Ms. right?

Simulations vs. Probability Models

1. Compare the similarities and differences between a prediction based on a simulation and one based on a mathematical model.
• Both require simplifying assumptions.

• Both require are excellent for answering what-if questions.

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.

1. 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.

1. 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.

 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

1. 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).

2. Model the expected decision making accuracy for:

• unanimous decisions
• 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

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.
 Essential Question: Are all disjointed  events independent?

Solving Probability Problems for Unions and intersections given independent or disjointed events

1. Draw the Venn diagram and define a union (A or B) or intersection (A and B) for a collection of events.

2. Calculate the probabilities for unions (unions correspond to or-statements) with independent and disjointed events.

Homefun (formative/summative assessment): 6.27, 6.31

Lesson 4
Key Concept: Probabilities with Unions
Purpose: Solve probability problems with both disjointed and non-disjointed independent events.
 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) Disjointed: 2 coins both heads or both tails Not Disjointed: 2 coins at least one head or at least one tail

Problem Solving (Teams of two): Work problems 6.39 and 6.40

 Essential Question: What are conditional probabilities?

Solving Conditional Probability Problems for Unions and intersections

1. Calculate conditional probabilities for unions ( contain or-statements, p.348-349).

2. Calculate conditional probabilities for intersections ( contain and-statements, p.348-349).

3. Determine if 2 events are independent using the test P(B|A) = P(B).

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) P(B|A) = P(A and B)/P(A) 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

 Essential Question: What are the dangers of massive drug abuse testing programs ?
Solving Probability Problems With Diagrams - Why you
should be a tree hugger
1. 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)

2. Use trees for probability calculations with dependent events. P(B|A) ≠ P(B)

3. 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.

• disjointed -- events are mutually exclusive.

• independent -- the occurrence of one event gives no additional information about the other.

• conditional -- the occurrence of one event gives additional information about the other.

1. 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.

 Essential Question: How can I make an "A" on the test?

Probability  Review

1. Master the vocabulary
2. Correctly draw tree diagrams and relate them to Venn diagrams.
3. Work the practice test.
4. 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

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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