Chapter 5 - Probability

DAY CLASSWORK HOMEWORK

0 Chapter 4 Test1) Read pg. 283 – 289 in the book2) Complete pg. 3 of the notes in here3) Do problems on p. 293 (#3, 9, 11, 12) in your book

1

2

3

4 Review

5 Chapter 5 Test Start Ch. 6

Lesson Objectives: Throughout this chapter we will focus on learning the rules of probability and how to use them to calculate the probability of various types of events. We will also learn the terminology of probability, such as, independent and dependent events, mutually exclusive (or disjoint) events, union, intersection, and joint probabilities. Probability calculations can be difficult for many students, so we will do many practice problems to gain more experience and in turn a better understanding of how to approach probability problems.

1

Chapter 5 Topics Page Number

5.1 - Simulations......................................................................................................4

5.2 – Probability Models..........................................................................................6

5.2 – Two-Way Tables, Venn Diagrams & The General Addition Rule...................8

5.3 – The General Multiplication Rule, Conditional Prob., Independence.............13

Probability Activity:

What is the probability that a thumb tack lands pointed end up? Is it 50/50?? When it is impossible to mathematically calculate the probability of an event or if you want to test a claimed probability, then we must explore the empirical probability, which is to say, the experimental probability to approximate the actual probability. We must trust in the law of averages in conjunction with the law of large numbers to get an approximation for the actual probability. So, with your thumb tack, toss it 50 times and record the proportion that it landed pointed end up. We will amass the class’ data to let the law of averages give us an approximate probability.

Proportion of times out of 50 tosses thumb tack landed pointed-end up for you: __________

Proportion of times out of 50 tosses thumb tack landed pointed-end up for the class: __________

Dotplot showing the class data:

_____________________________________________

1. Explain what the probability found by the entire class means in context to the definition of probability.

2. How could we have reduced the amount of variation we witnessed from person to person in the dotplot?

Now let’s look at a Probability Applet to explore the Law of Averages + the Law of Large Numbers once again.

http://www.rossmanchance.com/applets/randomBabies/Babies.html

2

Guided Notes for “Ideas of Probability” Start on page 283 in the book.

1. Why do statisticians recommend random samples and randomized experiments? _________________________________________________________________________________ _________________________________________________________________________________

2. What is the “big fact” of watching random events occur? ____________________________________________________________________________________________________________________

_________________________________________________________________________________

3. Read the example on “Tossing Coins”. As the number of coin tosses increased, the proportion of heads started to _______________________________________________________________________

4. Def: Law of Large Numbers _____________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________

5. Def: Probability _______________________________________________________________________ ______________________________________________________________________________________ ______________________________________________________________________________________

6. Probability gives us a language to describe ________________________________________________ ______________________________________________________________________________________

7. “Check your understanding” on pg. 286: #1a: _________________________________________________________________________________ _________________________________________________________________________________ #1b: _________________________________________________________________________________

_________________________________________________________________________________ #2a: __________ #2b: ___________ #2c: ___________ #2d: _____________

8. What is the “myth” about the law of averages? ____________________________________________________________________________________________________________________________

9. Read why a family won’t intentionally go for a boy after having three girls on pg. 289.

3

5.1 – Simulation

This section deals with questions involving chance. For example: Flipping a coin: What is the chance that we will get 3 heads in a row with 10 flips?

Children: A couple wants 4 kids, what is the chance they will have at least 1 girl?

Airlines: Should airlines overbook, and by how much, to account for no-shows?

There are 3 methods we can use to answer questions involving chance:

1. Try to __________ the likelihood of a result of interest by actually carrying out the ____________ many

times and calculating the result’s relative frequency.

2. Develop a ____________ ____________ and use it to calculate a theoretical answer.

3. Start with a model that, in some fashion, reflects the truth about the experiment, and then develop a

procedure for ______________ - or ________________ – a number of repetitions of the experiment.

The imitation of chance behavior, based on a model that accurately reflects the experiment under consideration, is called a Simulation.

How to perform a simulation:

a. State the problem or describe the experiment

b. State the assumptions

c. Assign digits to represent the outcomes

d. Simulate many repetitions

e. State your conclusions

Practice Problems:

1. How likely is a run of at least 3 consecutive heads or 3 consecutive tails in 10 tosses of a coin?

a. State the problem or describe the experiment

b. State the assumptions

c. Assign digits to represent the outcomes

d. Simulate many repetitions

4

e. State your conclusions

2. In the game of Scrabble, each player begins by drawing 7 tiles from a bag containing 100 tiles. There are 42 vowels, 56 consonants, and 2 blank tiles in the bag. About 3% of the time, the first player in Scrabble can “bingo” by playing all 7 tiles on the first turn. How many games of Scrabble would you expect to have to play, on average, for this to happen? Design and carry out a simulation to answer this question. a. State the problem or describe the experiment

b. State the assumptions

c. Assign digits to represent the outcomes

d. Simulate many repetitions

e. State your conclusions

Often, the hardest part of a simulation is assigning the digits to the outcomes. Let’s practice that!

1. Choose a person at random from a group of which 70% are employed.

2. Choose a person at random from a group of which 73% are employed.

3. Choose one person at random from a group of which 50% are employed, 20% is unemployed, and 30%

are not in the labor force.

5

4. Simulate the next ten frozen yogurt flavor orders given the following frequencies: 38% chocolate, 42%

vanilla, 20% strawberry

5.2 – Probability Models

Probability Models Include: A list of possible outcomes (Sample Space) and a probability for each outcome.

Vocabulary of Probability Models:1. Sample Space: _________________________________________________________________

2. Event: ________________________________________________________________________

______________________________________________________________________________

3. Probability Model: ______________________________________________________________

______________________________________________________________________________

Ex 1: Tossing a coin. S = { }

Outcome

Probability

Ex 2: The Sum of 2 Die Rolls S = { }

Ex 3: An experiment consists of spinning the spinner at the right.

List the sample space: ______________________________________

Ex 4: A couple plans to have 3 children with hopes of having 3 girls. How many outcomes are there in the sample space? _____________.

List the sample space. ___________________________________________________________

6

Outcome

Probability

0.5

0.75 0.25

0

Formal Probability Rules:1. The probability P(A) of any event A satisfies ________________________________.

2. If S is the sample space in a probability model, then ______________.

3. The ____________________of any event A is the event that A does not occur, written as ______. The

_________________________ states that ________________________.

Picture:

Complementary events are two ___________ events

whose union is the ______________________.

A + Ac = ______ A Ac = ________

A Ac = ________ This is read “A intersect Ac is empty”.

4. Two events A and B are ___________________ (or mutually exclusive) if they have no outcomes in

common (can never occur simultaneously). If A and B are disjoint, _____________________________.

This is the ___________________ for ________________ events.

Picture:

Disjoint events have ________________ in common,

Therefore, in the Venn diagram they have _________________.

The event {A B}, read “A union B” is another way to say A or B.

If A and B are disjoint then A B = _________

Ex 5: Marital Status of Young Women

Select a woman at random age 25 to 29 years old and record her marital status. The probability of any marital status is the proportion of all women aged 25 to 29 who have that status. Here is the probability model.

Marital Status Never Married Married Widowed DivorcedProbability .353 .574 .002 .071

1. Is probability rule #1 and 2 satisfied? Explain. ______________________________________________

2. What is the probability that a randomly selected woman is not married? __________________________

What rule was used to determine this probability? _____________________________

3. Find the probability that a randomly selected woman has never been married or is divorced? ________.

What rule was used to determine this probability? _____________________________________

What is the name for two events that have no outcomes in common? ___________________________

5.2 – General Addition Rule

7

Probability Rules:

1. _________________________________

2. _________________________________

3. _________________________________

4. _________________________________

An easy way to organize data involving two variables is through a two-way table (we did this in Ch. 1).

What is the probability of selecting a 25 to 34 year old? _______________

What is the probability of selecting a high school graduate? _____________

What is the probability of selecting a 25 to 34 year old or a high school graduate? (Hint: when we use the term

“or” in here, we do not just mean one OR the other, but rather one OR the other OR both!) ________________

Problem: The addition rule only works for two disjoint events. We need a better general addition rule that works for all situations, despite whether or not they are disjoint.

For two events, A and B, the union is the event ___________ meaning that A occurs, B occurs, or both occur.

Two or More Disjoint Events Non-Disjoint Events

8

Picture:

Rule:

Card Examples:

Ex 1: Deborah and Matthew are anxiously awaiting word on whether they have been made partners of their law firm. Deborah guesses that her probability of making partner is 0.7 and that Matthew’s is 0.5. Deborah also guesses that the probability that both she and Matthew are made partners is 0.3.

1. Let A = {Deb Promoted} and B = {Matt Promoted}. Draw a Venn diagram to illustrate this situation using the following in terms of A, B, AC, BC, and list all the probabilities.

2. Write a probability statement and calculate the probability that at least one is promoted.

3. Write a probability statement and calculate the probability that neither of them is promoted.

4. Write a probability statement and calculate the probability that Matt or Deb is promoted.

Key Terms:1. Disjoint (or Mutually Exclusive): No outcomes in common; no overlap in Venn Diagram

2. Intersection: What 2 events have in common; A and B; A B;

3. Union: Put Together; A or B; A B;

Quiz Worksheet 5.2

9

10

11

5.3 – The Multiplication Rule

Probability Rule #5:

Two events A and B are ___________________ if knowing that one occurs does not change the probability

that the other occurs. If A and B are independent, then __________________________________________.

This is the ______________________________________________________________________________.

Classify the following events as independent or dependent and find the probabilities for #1-3:

1. Draw an ace, replace it, shuffle, and then draw another ace. ___________________________

2. Draw a heart, shuffle, and then draw a diamond. ___________________________________

It is not uncommon for people to confuse the concepts of mutually exclusive events and independent events.

Mutually Exclusive Events: If event A happens, then event B cannot, or vice-versa.

Independent Events: The outcome of event A has no effect on the outcome of event B.

So, if A and B are mutually exclusive, they cannot be independent.If A and B are independent, they cannot be mutually exclusive.

When do you use the rule for disjoint events? _________________________________________________

When do you use the rule for independent events? ____________________________________________

Applying the Probability Rules:1. A transatlantic telephone cable contains repeaters at regular intervals to amplify the signal. If a repeater

fails, it must be replaced by fishing the cable to the surface at great expense. Each repeater has probability 0.999 of functioning without failure for 10 years. Repeaters fail independently of each other. (This assumption means that there are no “common causes” such as earthquakes that would affect several repeaters at once).

a. Find the probability that two repeaters function properly for 10 years.

b. Find the probability that 10 repeaters function property for 10 years.

c. The transatlantic cable has 300 repeaters. Find the probability that they all work properly for 10 years.

12

2. The letters of the word STATISTICS are placed on index cards and thrown in a bag.

a. List the Sample Space. ___________________________________

b. Construct the probability distribution:

Outcome

Probability

c. Let V ={Vowel} , and M ={Letter falls in the first half of the alphabet, between A and M}. List the sample space for each of the following and find their probabilities.

V = { _________ } P(V) = _________

M = { _________ } P(M) = __________

V or M = { _________ } P(V or M) = __________

MC = { ___________ } P(Mc) = __________

3. The Daily Number (Big 3) has been rigged! The even numbered balls have been made lighter so that they occur 3 times as often as the odd numbered balls, which are equally likely to occur.

a. Construct the probability distribution:

OutcomeProbability

b. Let A = {odd number drawn}. Find the sample space of A and P(A).

c. Let B = {prime number drawn}. Find the sample space of B and P(B).

d. Are events A and B disjoint? Explain.

13

Probability Rules:

1. _________________________________

2. _________________________________

3. _________________________________

4. _________________________________

5. _________________________________

Problem: The multiplication rule only works for independent events. We need a more general multiplication

rule that works for all situations, despite whether or not the events are independent.

General Multiplication Rule:The joint probability that both of two events A and B happen together can be found by the following rule:

P(A and B) = P(A)●P(B|A)

Here P(B|A) is the ____________________ probability that B occurs given the information that A occurs.

P(A and B) = P(A)●P(B|A)

Rewrite the equation above, solving the conditional probability: __________________________________

Note:

Two events are independent if ___________________________________________________________

___________________________________________________________________________________.

Stated Mathematically: Two events are independent if _____________________________________ or

if __________________________________________.

14

Ex 1: Slim wants very much to draw 2 diamonds in a row. As he looks at his hand and at the upturned cards on the table, Slim sees 11 cards. Of these, 4 are diamonds.

a. Draw a tree diagram for this situation complete with probabilities.

b. Find probability that Slim draws 2 diamonds in a row.

P(1st card diamond AND 2nd card diamond) =

c. Find the probability that he gets exactly 1 diamond out of the two cards.

Ex 2: Use the table below of Age and Marital Status (thousands of women) to find the following probabilities. Identify each probability as either joint or conditional:

15

Age

18 to 24 25 to 64 65 and over Total

Married 3,046 48,116 7,767 58,929

Never Married 9,289 9,252 768 19,309

Widowed 19 2,425 8,636 11,080

Divorced 260 8,916 1,091 10,267

Total 12,614 68,709 18,262 99,585

1. P (married) = ________________________ Joint or Conditional?

2. P (married | age 18 to 24) = ________________________ Joint or Conditional?

3. P (married and age 18 to 24) = ________________________ Joint or Conditional?

4. P (age 18-24 | never married) = ________________________ Joint or Conditional?

5. P (never married and age 65 and over) = ________________________ Joint or Conditional?

6. Are the events A and B independent? Explain.

AP Statistics – Testing for Independence

16

Determine which table contains independent events. Show mathematical calculations and a written explanation to prove your answer.

17

5.3 Review

18

19

Practice Worksheet A

A couple plans to have three children. Find the probability that the children are

1. all boys

2. all girls

3. two boys or two girls

4. at least one child of each sex.

In a statistics class there are 18 juniors and 10 seniors; 6 of the seniors are females, and 12 of the juniors are males. If a student is selected at random, find the probability of selecting

20

5. a junior or a female

21

6. a senior or a female

7. a junior or a senior

Practice Worksheet BIn building new homes, a contractor finds that the probability of a homebuyer selecting a two-car garage is 0.70 and of selecting a one-car garage is 0.20. (Note that the builder will not build a three-car or larger garage.)

1. What is the probability that the buyer will select either a one-car or a two-car garage?

2. Find the probability that the buyer will select no garage.

3. Find the probability that the buyer will not want a two-car garage.

22

Practice Worksheet C

Here are the counts (in thousands) of earned degrees in the United States in a recent year, classified by level and by the sex of the degree recipient:

Bachelor’s Master’s Professional Doctorate TotalFemale 616 194 30 16 856

Male 529 171 44 26 770Total 1145 365 74 42 1626

1. If you choose a degree recipient at random, what is the probability that the person you choose is a woman?

2. What is the probability that a randomly chosen degree recipient is a man?

3. What is the conditional probability that you choose a woman, given that the person chosen received a professional degree?

4. What is the conditional probability that the person chosen received a bachelor’s degree, given that he is a man?

5. Are the events “choose a woman” and “choose a professional degree recipient” independent? How do you know?

6. Use the multiplication rule to find the probability of choosing a male bachelor’s degree recipient.

23

7. Confirm your answer to #6 by finding the probability of choosing a male bachelor’s degree recipient directly from the table of counts above.

Practice Worksheet A Answers:1. 1/8 Note: The sample space is: BBB,2. 1/8 BGG, GBG, GGB3. 3/4 GBB, BGB, BBG4. 3/4 GGG5. 24/28 which reduces to 6/76. 16/28 which reduces to 4/77. 28/28 which is 1 or 100% chance.Note: There are 18 Juniors, 12 of which are male, so 6 are female.

There are 10 Seniors, 4 of which are male, because 6 are female.

Practice Worksheet B Answers:1. 0.2 + 0.7 = 0.9. or a 90% chance the buyer will select a 1 or 2 car garage.2. Since 90% of people will want a 1or 2-car garage, which means 10% will not want a garage.3. 70% of people want a 2-car garage, so 30% of people do not want a two-car garage.

Practice Worksheet C Answers:1. 856 / 1626 = 52.64%2. 770 / 1626 = 47.36% (or you can subtract the previous answer from 100%)3. 30 / 74 = 40.54%4. 529 / 770 = 68.70%5. Let A = Choose a woman and B = Choose a professional degree recipient. If they are independent then

they would satisfy the multiplication rule for independent events: P(A and B) = P(A) P(B)P(A and B) = P(woman and Professional degree recipient) = 30 / 1626 = .01845P(A) = 856 / 1626 = .5264P(B) = 74 / 1626 = .0455Substituting these values into the equation you get: 0.01845 = (0.5264)(0.0455)

Since 0.01845 0.024, the events are not independent

24

6. Let A = male and B = Bachelor Degree. We are asked to find P(A and B) with the following rule.P(A and B) = P(A) P(B | A) = (770 / 1626) (529 / 770) = 0.3253 = 32.53%

7. To find the probability of choose a male bachelor degree recipient directly from the table I divide the number of male bachelor recipients (529) by the table total (1626) to get 32.53%.

Preparing for Your Chapter 5 Test

For the Chapter 5 Test You Should Know:

The definition and examples of disjoint, independent, dependent, and complementary events

like the back of your hand.

How to verify if a probability distribution is legitimate by seeing if 0 P(A) 1 and P(S) = 1

How to answer various probability questions using the multiplication rule for independent

events.

How to answer various probability questions using the addition rule (for disjoint events as well

as the general rule for non-disjoint events).

How to assign probabilities to equally likely events and unequally likely events (such as the

“fixed” lottery example.

How to find the total number of outcomes either with the multiplication principle or by writing

out the entire sample space. You should know how many outcomes there are when you roll

two die, flip 2, 3, 4, or 5 coins, and when you draw cards…by heart!

How to determine if two events are independent by seeing if P(A and B) = P(A)P(B).

How to construct and label a Venn Diagram in terms of A, B, Ac, and Bc. How to answer

probability questions based upon your Venn Diagram

That P(at least one _____) = 1 – P(No _____). For example, P(at least one girl among 4 kids)

= 1 – P (no girls among 4 kids)

That the following two versions of the general multiplication rule are interchangeable: P(A and

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

How to draw a tree diagram and calculate probabilities from it.

25

Chapter 5 – Important Terms

1. Random

2. Probability

3. Probability Model

4. Sample Space

5. Tree Diagram

6. Multiplication Principle

7. With Replacement

8. Without Replacement

9. Event

10. 5 Probability Rules

a. 0 P(A) 1 Probabilities must fall between 0 and 1

b. P(S) = 1 The Probability of the Sample Space is 1

c. P(Ac) = 1 - P(A) Complement Rule

d. P(A or B) = P(A) + P(B) – P(A and B) General Addition Rule

e. P(A and B) = P(A) P(B | A) General Multiplication Rule

11. Independent Events

12. Dependent Events

26

13. Disjoint Events

14. Complementary Events

15. Intersection

16. Union

27