Unit 5 – Probability

Unit 5 - Probability (AP Stats)

Unit 5

Warm-Ups

Unit 5

Reading

Guide

TPS: Review

pg 276 #1 - 17

TPS: 5.1

Pg 293 #1 – 25

odds, 31 – 36

all

TPS: 5.2

Pg 309 #39 –

55 odds, 57 –

60 all

TPS: 5.3

Pg 329 # 63 –

89 odds, 101,

104 – 106 all

TPS Review Work

(optional) Pg. 334 #1 – 11

Pg. 336 # 1 - 14

Duck Hunter

Activity

Name _______________

Score _________

(write our quotes on the back)

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Daily Plan for Unit 5 - Probability

Day 1

* Day 1 - Simulation Activity wkst

* Cumulative AP Practice Test TPS

* Introduce GREED game

TPS: pg 276 #1 - 17

Day 2

* GREED warm up

* Intro to Prob. Video (AAO)

* 5.1 “Flipping Coins, Making

Predictions, and Shooting Hoops”

Simulation wkst

TPS: 5.1 Exercises

Pg 293 #1 – 25 odds, 31 – 36 all

Day 3 * GREED warm up

* 5.2 Probability Notesheet

* Prob. Models video (AOO)

TPS: 5.2 Exercises

Pg 309 #39 – 55 odds, 57 – 60 all

Day 4 * GREED warm up

* 5.3 Conditional Probability Notesheet

TPS: 5.3

Pg 329 # 63 – 89 odds, 101, 104 – 106 all

Day 5 Duck Hunters Activity

* Work on Unit Review Homework

TPS Review Work (optional)

Pg. 334 #1 – 11

Pg. 336 # 1 - 14

Day 6

Unit 5 TEST

Day 1 - Probability Simulations

1 in 6 WINS! As a special promotion for its 20-ounce bottles of soda, a soft drink company printed a message on the

inside of each bottle cap. Some of the caps said, “Please try again!” while others said, “You’re a winner!”

The company advertised the promotion with the slogan “1 in 6 wins a prize.” Seven friends each buy one

20-ounce bottle at a local convenience store. The store clerk is surprised when three of them win a prize. Is

this group of friends just lucky, or is the company’s 1-in-6 claim inaccurate? In this Activity, you and your

classmates will perform a simulation to help answer this question.

Let’s assume that the company is telling the truth, and that every 20-ounce bottle of soda it fills has a 1-in-6

chance of getting a cap that says, “You’re a winner!” We can model the status of an individual bottle with a

six-sided die: let 1 through 5 represent “Please try again!” and 6 represent “You’re a winner!”

1. Roll your die seven times to imitate the process of the seven friends buying their sodas. How many

of them won a prize? Record your results in the table – CIRCLE any winning rolls.

2. Repeat Step 1 four more times. In your five repetitions of this simulation, how many times did three

or more of the group win a prize?

3. Combine results with your classmates. What percent of the time did the friends come away with

three or more prizes, just by chance?

# of times 3 or more prizes = _________ # of trials = __________ % = ___________

4. Based on your answer in Step 3, does it seem plausible that the company is telling the truth, but that

the seven friends just got lucky? Write down your thoughts.

Roll # 1 2 3 4 5 6 7

Result

Roll # 1 2 3 4 5 6 7

Result

Roll # 1 2 3 4 5 6 7

Result

Roll # 1 2 3 4 5 6 7

Result

Roll # 1 2 3 4 5 6 7

Result

Airling Overbooking

Mudlark Airlines has a 15-seater commuter turboprop that is used for short flights. Their data suggest that

about 8% of the customers who buy tickets are no-shows. Wanting to avoid empty seats and avoid missing

an opportunity to increase revenue, they decide to sell 17 tickets for each flight. Ticketed customers who

can’t be seated on the plane will be accommodated on another flight and will receive a certificate good for

a free flight at another time. Design and carry out a couple different simulations to estimate the probability

that at least one ticket-holder is denied a seat on the plane if 17 tickets are sold.

A) How could you simulate this situation using your calculator? Carry out the simulation.

B) How could you simulate this situation using the Table of Random Digits? Carry out the simulation.

“Flipping Coins, Making Predictions and Shooting Hoops”

Simulation #1:

Situation: You are about to take a True/False quiz that you have not studied for. The quiz will consist of 20

questions and you have decided that you have a better chance of doing well on the quiz by simply

guessing. Here is your answer sheet – fill in the blanks with either T or F and let’s see how you do.

1) _____ 6) _____ 11) _____ 16) _____

2) _____ 7) _____ 12) _____ 17) _____

3) _____ 8) _____ 13) _____ 18) _____

4) _____ 9) _____ 14) _____ 19) _____

5) _____ 10) _____ 15) _____ 20) _____

The answers will appear on the board – how many did you get correct? ___________

1) If this was a multiple choice quiz, with 4 possible answers for each question, do you think you would do

better or worse? By how much?

2) Look back at your answers; do you think that your guesses would constitute as being “random”?

Why or why not?

3) Using a fair coin, toss it in the air 20 times and let it fall to the ground. Record your results below.

1) _____ 6) _____ 11) _____ 16) _____

2) _____ 7) _____ 12) _____ 17) _____

3) _____ 8) _____ 13) _____ 18) _____

4) _____ 9) _____ 14) _____ 19) _____

5) _____ 10) _____ 15) _____ 20) _____

4) If Heads = False and Tails = True, how many of the questions would you have gotten correct? _______

5) Now, using your calculator and randomness, get your answers for this quiz. (Show how you set this

up on your calculator)

1) _____ 6) _____ 11) _____ 16) _____

2) _____ 7) _____ 12) _____ 17) _____

3) _____ 8) _____ 13) _____ 18) _____

4) _____ 9) _____ 14) _____ 19) _____

5) _____ 10) _____ 15) _____ 20) _____

* Now how many did you get correct? _____________

6) How many of the 20 questions would “expect” to get correct by using a random selection of Trues

and Falses? ___________

7) Look back at your three sets of solutions, how close to your number from #6 did you come?

PART II

8) Now, look back at your random selection of answer for #5 and count how many runs of 2, 3, 4 or

more you had of any answer (for instance, TTTFTFT – this has a run of 3 Trues).

9) How many runs of 2, 3, 4 or more did you have in your first selections? What do you notice?

** The MYTH of SHORT RUN REGULARITY **

** The MYTH of the LAW OF AVERAGES **

Simulation #2

It was Jan. 23, 2015, against the Sacramento Kings, and Thompson was ice cold coming out of halftime. The

then-24-year-old had missed his previous five shots, including an uncontested layup from point-blank range,

and a pair of 3-pointers without a defender in the vicinity. No one could have foreseen what happened next.

With the score tied with 9 minutes, 45 seconds left in the third quarter, Thompson's masterpiece began.

Over the next 10 minutes of action, Thompson scored 37 points on 13 consecutive made shots – 9 three-

pointers, 4 two-pointers and 2 free-throws. Everyone watching the game knew Thompson had the “hot

hand” but some statisticians disagree. Read over some of the following articles and see what you think.

http://www.espn.com/nba/story/_/page/presents-19573519/heating-fire-klay-thompson-truth-hot-

hand-nba

http://www.nytimes.com/2008/03//30/opinion/30strogatz.html

What are your thoughts?

Notes on Probability

Probability: The probability of any outcome of a chance process is a number between _____ (never

occurs) and _______ (always occurs) that describes the p_________________ of times the outcome would

occur in a very long series of r__________________.

A S__________________ is the imitation of chance behavior, based on a model that accurately reflects the

situation. Here are the steps in performing a simulation:

State: What is the Q______________ of I__________________ about some chance process?

Plan: Describe how to use a C_____________ D____________ to imitate one repetition of the process.

Explain clearly how to identify the O_________ of the chance process and what V____________ to

measure.

Do: Perform many R__________________ of the simulation.

Conclude: Use the R_______________ of your simulation to answer the question of interest.

Probability Models

Sample Space: The set of _____________________________________________.

Probability Model: A description of some chance process that consists of _____ parts: a S__________

____________(S) and a P_________________ for each outcome.

Ex. Construct a Probability Model for rolling a pair of dice:

Events are C_________________ of outcomes from some chance process. That is, an event is a S___________

of the sample space. Events are usually designated by capital letters, like A, B, C, and so on.

Ex. Let A = sum of 5; B = sum of 12; and C = sum is not 7 or 11. Find:

P(A) = P(B) = P(C) =

Probability RULES:_

The probability of any event is a number between _______ and _______

All possible outcomes together must have probabilities whose S_______ = 1.

If all outcomes in the sample space are equally likely, the probability that event A occurs can be

found using the formula:

The probability that an event does not occur is __________ the probability that the event does occur.

If two events have no outcomes in common, the probability that one or the other occurs is the sum

of their I__________________ P___________________.

Mutually Exclusive: When two events have _____ _______________ in common and so can never occur

together. (Also know as D______________ events.)

• For any event A, ____ ≤ P(A) ≤ _____.

• If S is the sample space in a probability model, P(S) = ___

• In the case of equally likely outcomes, P(A) =

• Complement rule: P(AC) = ______ – P(A)

• Addition rule for mutually exclusive events: If A and B are mutually exclusive,

P(A or B) = ___________________________

Ex. Distance-learning courses are rapidly gaining popularity among college students. Randomly select an

undergraduate student who is taking distance-learning courses for credit and record the student’s age.

Here is the probability model:

(a) Show that this is a legitimate probability model.

(b) Find the probability that the chosen student is not in the traditional college age group (18 to

23 years).

(c) Find the probability that the chosen student is between 24 and 39 years old.

Two-Way Tables and Probability: When finding probabilities involving two events, a two-way table can

display the sample space in a way that makes probability calculations easier.

Find the Probability that the student…

(Define events A: is male and B: has pierced ears.)

** Note, the previous example illustrates the fact that we can’t use the addition rule for mutually exclusive

events unless the events have no outcomes in common.

General Addition Rule for Two Events

If A and B are any two events resulting from some chance process, then

P(A or B) = _____________________________

Venn Diagrams

The Complement of A:

The intersection of events A and B (A ∩ B) is the set of all outcomes in both events A and B.

The union of events A and B (A ∪ B) is the set of all outcomes in either event A or B.

In the intersection of two circles Male and pierced ears ___________ ______

Inside circle A, outside circle B Male and no pierced ears ___________ ______

Inside circle B, outside circle A Female and pierced ears ___________ ______

Outside both circles Female and no pierced ears ___________ ______

Practice:

1. Suppose you choose a random U.S. resident over the age of 25. The table below is a probability model

for the education level the selected person has attained, based on data from the American Community

Survey from 2006-2008.

(a) What is the probability that a randomly selected person has a graduate or professional degree?

(That is, fill in the space marked with a “?”) Show your work.

(b) What is the probability that a randomly-selected person has at least a high school diploma? Show

your work.

2. There are 35 students in Ms. Ortiz’s Precalculus class. One day, 24 students turned in their homework

and 14 turned in test corrections. Eight of these students turned in both homework and test corrections.

Suppose we randomly select a student from the class.

(a) Fill in the Venn diagram below so that it describes the chance process involved here. Let H = the

event “turned in homework” and C = the event “turned in corrections.”

(b) What is the probability that a randomly-chosen

student turned in neither homework nor corrections?

Justify you answer with appropriate calculations.

3. Below is a two-way table that describes responses of 120 subjects to a survey in which they were asked,

“Do you exercise for at least 30 minutes four or more times per week?” and “What kind of vehicle do you

drive?”

Suppose one person from this sample is randomly selected.

(a) List two mutually exclusive events for this chance process.

(b) What is the probability that the person selected drives an SUV?

(c) What is the probability that the person selected drives either a sedan or a truck?

(d) What is the probability that the person selected drives a truck or exercises four or more times

per week?

Notes on Conditional Probability

Conditional Probability: The probability that one event happens given that another event is ___________

known to have happened.

*Suppose we know that event A has happened. Then the probability that event

B happens given that event A has happened is denoted by _________.

Ex. In a classroom of 30 students, in which there are 18 girls and 12 boys, find the conditional probability that

when all their names are thrown in a hat a girl’s name gets chosen given a girl’s name was already chosen on

the first draw. P(girl’s name │girl’s name)

Ex. E: the grade comes from an EPS course

L: the grade is lower than a B Find P(L)

Find P(E | L)

Find P(L | E)

Independent Events occur when one event has ____ ___________ on the chance that the other event will

happen. In other words, events A and B are independent if :

P(A | B) = __________ and P(B | A) = ____________

Ex. Are the events “male” and “left-handed” independent?

P(left handed │ male) =

P(left handed) =

Ex. Are the events “female” and “allergies” independent?

Female Male Total P(allergies │ female) =

Allergies 10 8 18

No Allergies 13 9 22 P(allergies) =

Total 23 17 40

Tree Diagrams

Draw a tree diagram for rolling a 6-sided die and flipping a coin.

Die Roll Coin Flip

1

2

3

4

5

6

Multiplication Rule of Probability

The probability that events A and B both occur can be found using the general multiplication rule

P(A ∩ B) = ________________

where P(B |A) is the conditional probability that event B occurs given that event A has already occurred.

Ex. What percent of all adult Internet users visit video-sharing sites?

Multiplication rule for INDEPENDENT EVENTS:

If A and B are independent events, then the probability that A and B both occur is: P(A ∩ B) =

Ex. Following the Space Shuttle Challenger disaster, it was determined that the failure of O-ring joints in

the shuttle’s booster rockets was to blame. Under cold conditions, it was estimated that the probability

that an individual O-ring joint would function properly was 0.977. Assuming O-ring joints succeed or fail

independently, what is the probability all six would function properly?

P(joint1 OK and joint 2 OK and joint 3 OK and joint 4 OK and joint 5 OK and joint 6 OK) =

If we rearrange the terms in the general multiplication rule, we can get a formula for the

conditional probability P(B | A).

General Multiplication Rule

Conditional Probability Rule

Ex. What is the probability that a randomly selected resident who reads USA Today also reads the

New York Times?

Practice:

1. Ivy conducted a taste test for four different brands of chocolate chip cookies. Below is a two-way table

that describes which cookie each subject preferred and their gender.

Suppose one subject from this experiment is selected at random.

(a) Find the probability that the selected subject preferred Brand C.

(b) Find the probability that the selected subject preferred Brand C, given that she is female.

(c) Are the events “preferred Brand C” and “female” independent? Explain.

(d) Are the events “preferred Brand C” and “female” mutually exclusive? Explain.

(e) If a random sample of two subjects is selected, what is the probability that neither preferred

Brand A?

2. Officials at Dipstick College are interested in the relationship between participation in (interscholastic)

sports and graduation rate. The following table summarizes the probabilities of several events when a

male Dipstick student is randomly selected.

(a) Find the probability that a student graduates, given that he participates in sports.

(b) Find the probability that the individual does not graduate, given that he participates in sports.

(c) Draw a tree diagram to summarize the given probabilities and those you determined above.

(d) Find the probability that the individual does not participate in sports, given that he graduates.