Chapter 4

Probability and Counting Rules

Section 4-1

Sample Spaces and Probability

Learning Target

• Determine sample spaces and find the probability of an event using classical probability or empirical probability.

Basic Concepts

• Probability Experiment – a chance process that leads to well-defined results called outcomes

• Outcome – the result of a single trial of a probability experiment

• Sample Space – the set of all possible outcomes of a probability experiment– Examples: tossing a coin (head, tail), roll a die

(1,2,3,4,5,6), Answer a true/false question (true, false)

Sample Space for Rolling Two Dice

Die 1Die 2

1 2 3 4 5 6

1 (1,1) (2,1) (3,1) (4,1) (5,1) (6,1)

2 (1,2) (2,2) (3,2) (4,2) (5,2) (6,2)

3 (1,3) (2,3) (3,3) (4,3) (5,3) (6,3)

4 (1,4) (2,4) (3,4) (4,4) (5,4) (6,4)

5 (1,5) (2,5) (3,5) (4,5) (5,5) (6,5)

6 (1,6) (2,6) (3,6) (4,6) (5,6) (6,6)

Cards in a Regular Deck of Cards

• 4 suits – spades, diamonds, hearts, clubs• 13 of each suit• There are 3 face cards in each suit – jack

queen, king• 52 cards total

Gender of Children

• Find the sample space for the gender of the children if a family has three children.

• How can this be done?

Tree diagram

• Device consisting of line segments emanating from a starting point and also from the outcome point. It is used to determine all possible outcomes of a probability experiment.

B

G

1st child 2nd child 3rd child

Outcomes

BBB

BBG

BGB

BGG

GBB

GBG

GGB

GGG

More Vocab

• Event – consists of a set of outcomes of a probability experiment

• Simple Event – an event with one outcome• Compound Event – event with more than one

outcome– Example: the event of rolling an odd number on a

die

Classical Probability

• Uses sample spaces to determine the numerical probability that an event will happen

• Assumes that all outcomes are equally likely to occur

• Formula – number of outcomes in E divided by the total number of outcomes in the sample space,

• Probabilities can be expressed as fractions, decimals, or percents. Most problems will be expressed as fractions or decimals. If the problems starts in fractions the answer should be a fraction. If the problem starts as a decimal the answer should be a decimal.

• Fractions should always be reduced and decimals rounded to two or three decimal places.

Practice Problems

• A card is drawn at random from an ordinary deck of cards. Find these probabilities.– Of getting a jack– Of getting a red ace– Of getting the 6 of clubs– Of getting a 3 or a diamond– Of getting a 3 or a 6

• If a family has three children, what is the probability that two of the three children are girls?

Solutions

4 Probability Rules

1. The probability of any event E is a number (either a fraction or decimal) between and including 0 and 1. This is denoted by

2. If an event E cannot occur (i.e., the event contains no members in the sample space), its probability is 0.

3. If an event E is certain, its probability is 1. 4. The sum of the probabilities of all the

outcomes in the sample space is 1.

Complementary Events

• The complement of an event E is the set of all outcomes in the sample space that are not included in the outcomes of event E. The complement of E is denoted by E (read “E bar”)

• Example: The event E of getting an odd number is 1,3,5. The complement of E is getting an even number (2,4,6).

Practice Problems

• Find the complement of each event.– Rolling a die and getting a 4– Selecting a letter of the alphabet and getting a

vowel– Selecting a month and getting a month that begins

with a J– Selecting a day of the week and getting a weekday

Solutions

• Getting a 1,2,3,5,6• Getting a consonant (assume y is a consonant)• Getting February, March, April, May, August,

September, October, November, or December• Getting Saturday or Sunday

Rule for Complementary Events

• or or

Empirical Probability

• Relies on actual experience to determine the likelihood of outcomes

• Given a frequency distribution, the probability of an event being in a given class is

Practice Problems

• In a sample of 50 people, 21 had type O blood, 22 had type A blood, 5 had type B blood, and 2 had type AB blood. Set up a frequency distribution and find the following probabilities. – A person has type O blood– A person has type A or type B blood– A person has neither type A nor type O blood– A person does not have type AB blood

Solution

A. B. C. D.

Type FrequencyA 22B 5

AB 2O 21

Total = 50

Applying the Concepts and Exercise 4-1

Section 4-2

The Addition Rules for Probability

Learning Target

• IWBAT find the probability of compound events, using the addition rules.

Mutually Exclusive

• Two events are mutually exclusive if they cannot happen at the same time.

• In other words they have no outcomes in common.

• Example: getting a 4 and a 6 are mutually exclusive.

Which ones are mutually exclusive?

a. Getting an odd number and getting an even number

b. Getting a 3 and getting an odd numberc. Getting a 7 and a jackd. Getting a club and getting a king

Addition Rule #1

•When two events are mutually exclusive, the probability that A or B will occur is

Practice Problems

• A box contains 3 glazed doughnuts, 4 jelly doughnuts, and 5 chocolate doughnuts. If a person selects a doughnut at random, find the probability that either is a glazed or chocolate doughnut.

• At a political rally, there are 20 republicans, 13 democrats, and 6 independents. If a person is selected at random, find the probability that he or she is either a democrat or an independent.

Answers

Addition Rule #2

• If A and B are not mutually exclusive, then

Practice Problems

• In a hospital unit there are 8 nurses and 5 physicians; 7 nurses and 3 physicians are females. If a staff person is selected, find the probability that the subject is a nurse or a male.

AnswerStaff Females Males Total

Nurses 7 1 8

Physicians 3 2 5

Total 10 3 13

𝑃 (𝑛𝑢𝑟𝑠𝑒𝑜𝑟 𝑚𝑎𝑙𝑒)=𝑃 (𝑛𝑢𝑟𝑠𝑒 )+𝑃 (𝑚𝑎𝑙𝑒 )−𝑃 (𝑛𝑢𝑟𝑠𝑒𝑎𝑛𝑑𝑚𝑎𝑙𝑒 )= 813

+313−113

=1013

For 3 events

• Mutually Exclusive

• Not Mutually Exclusive

Venn Diagrams

Mutually Exclusive

Not Mutually Exclusive

P(A) P(B)

P(A) P(B)

P(A and B)

Exercises 4-2 1-25 odd and #8

• Graded for correct answer– 2, 6, 10, 14, 18, 20, 24, 26

Section 4-3

The Multiplication Rules and Conditional Probability

Learning Target

• IWBAT find the probability of compound events, using the multiplication rule.

Multiplication Rules

• The multiplication rules are used to find the probability of events that happen in sequence.

• For example, when you toss a coin and roll a die, you can find the probability of flipping a head and rolling a 4.

• The events are independent since the outcome of the first event does not effect the second.

Rule #1

• When two events are independent, the probability of both occurring is

Examples

• A coin is flipped and a die is rolled. Find the probability of getting a head on the coin and a 4 on the die.

• A card is drawn from a deck and replaced; then a second card is drawn. Find the probability of getting a queen then an ace.

More Examples

• An urn contains 3 red marbles, 2 blue marbles, and 5 white marbles. A marble is selected and its color noted. Then it is replaced. A second ball is selected and its color noted. Find the probability of each of these. – Selecting 2 blue marbles– Selecting 1 blue marble then 1 white marble– Selecting 1 red marble then 1 white marble

Solutions

The multiplication rule can be extended to three or more events by using the formula…

Dependent Events

When the outcome or occurrence of the first event effects the outcome or the occurrence of the second event in such a way that a probability is changed, the events are said to be dependent. When situations involve not replacing the item that was selected first, the events are dependent.

Conditional Probability

• The conditional probability of an event B in relationship to an event A is the probability that event B occurs after event A has already occurred.

Rule #2

• When two events are dependent, the probability of both occurring is

Means that B happens given that A happened first. (Conditional Probability)

Examples

• Three cards are drawn from an ordinary deck and not replaced. Find the probability of these events.– Getting 3 jacks– Getting an ace, a king, and a queen in order– Getting a club, a spade, and a heart in order– Getting 3 clubs

Solutions

Finding Conditional Probability

The probability that the second event B occurs given that the first event A has occurred can be found by dividing the probability that both events occurred by the probability that the first event has occurred. The formula is

Example

• A box contains black chips and white chips. A person selects 2 chips without replacement. If the probability of selecting a black chip and a white chip is , and the probability of selecting a black chip on the first draw is , find the probability of selecting the white chip on the second draw, given that the first chip selected was a black chip.

Solution

𝑃 (𝑊 𝐵 )= 𝑃 (𝐵𝑎𝑛𝑑𝑊 )𝑃 (𝐵)

=15/563/ 8

=1556÷38=1556∙83=57

Example 2• A recent survey asked 100 people if they thought women

in the armed forces should be permitted to participate in combat. The results of the survey are shown.

• Find the probability – The respondent answered yes, given that the respondent

was female.– The respondent was a male, given that the respondent

answered no.

Gender Yes No TotalMale 32 18 50

Female 8 42 50Total 40 60 100

Solutions

• Before you start, determine a variable to represent each outcome.

Let M=respondent was maleF=respondent was femaleY=respondent answered yesN=respondent answered no

a. Respondent answered yes given that the respondent was female.

The problem is to find P(Y F). The rule states . P(F and Y) is . P(F) is .Then P(Y F)=

b. Respondent was male, given the respondent answered no.

The problem is to find P(M N).

Probabilities for “at least”

• The multiplication rule can be used along with the complement rule to simplify problems involving “at least”.

Example #1

• A game is played by drawing 4 cards from an ordinary deck and replacing each card after it is drawn. Find the probability that at least one ace is drawn.

• Solution: Since the outcome is looking for 1 ace, 2 aces, 3 aces, or 4 aces, it would be easier to find the probability that 0 aces have been drawn and then subtracting from 1.

Example #2• A coin is tossed five times. Find the probability of

getting at least one tail.

P(at least 1 tail) = 1 – P(all heads)

Applying the Concepts and Exercises 4-3

Section 4-4

Counting Rules

Learning Target

• IWBAT find the total number of outcomes in a sequence of events, using the fundamental counting rule.

• IWBAT find the number of ways that r objects can be selected from n objects, using the permutation rule.

• IWBAT find the number of ways that r objects can be selected from n objects without regard to order, using the combination rule.

The Fundamental Counting Rule

• In a sequence of n events in which the first one has possibilities and the second event has possibilities and the third has and so forth, the total number of possibilities in the sequence will be

• These problems will include repetitions and no repetitions.

Example • A paint manufacturer wishes to manufacture

several different paints. The categories include

• How many different kinds of paint can be made if you can select one color, one type, one texture, and one use?

• Solution:

Color Red, blue, white, black, green, brown, yellow

Type Latex, oil

Texture Flat, semigloss, high gloss

Use Outdoor, indoor

Factorial Notation

• For any counting n

Example:

Note: You can find the factorial key under math PRB 4

Permutation

• A permutation is an arrangement of n objects in a specific order.

• The arrangement of n objects in a specific order using r objects is called a permutation of n objects taking r objects at a time. It is written as and the formula is

Example

• A television news director wants to use 3 news stories on an evening show. One story will be the lead story, one will be the second story and the last will be the closing story. If the director has a total of 8 stories to choose from, how many possible ways can the program be set up?

• Solution: • Note: You can use your calculator to do the

calculations. Type in number for n, then Math PRB nPr, then number for r.

Example 2

• A school musical director can select 2 musical plays to present next year. One will be presented in the fall, the other in the spring. If she has 9 to pick from, how many different possibilities are there?

• Solution:

Combinations

• A selection of distinct objects without regard to order is called a combination.

• The number of combinations of r objects selected from n objects is denoted by nCr and is given by the formula

Example

• A newspaper editor has received 8 books to review. He decides that he can use 3 reviews in his newspaper. How many different ways can these 3 reviews be selected?

• Solution: • Note: You can use your calculator for

calculations. Type in the number for n then math PRB nCr, then the number for r.

Example 2

• In a club there are 7 women and 5 men. A committee of 3 women and 2 men is to be chosen. How many different possibilities are there?

• Solution: • Note: Because there are two events and they

have to occur together you must multiply the events. Therefore, you will have to use rules from previous sections.

Exercises 4-4

1-47 odd

Section 4-5

Probability and Counting Rules

Learning Target

• IWBAT find the probability of an event using the counting rules.

Example 1Find the probability of getting four aces when five cards are drawn from an ordinary deck of cards.Steps: 1. Find the total number of ways to draw 5 cards

from a deck. (52C5 = 2598960)

2. Find the number of ways to draw 4 aces and 1 other card. (1 way to get four aces then 48 ways to get the other card, therefore there are 48 ways)

3. Find the probability of drawing 4 aces if 5 cards are drawn. (

Example 2A box contains 24 transistors, 4 of which are defective. If 4 are sold at random, find the following probabilities. a. Exactly 2 are defectiveb. None are defectivec. All are defectived. At least one is defective

A. Exactly 2 are defective

Steps:1. Find the number of ways we can draw 4

transistors. (24C4=10626)

2. Find the ways that 2 are defective and 2 are not defective. (4C2 and 20C2)

B. None are defectiveSteps: 1. Find number of ways to draw 4 transistors

(10626)2. Find number of ways to choose no defective.

(20C4)

C. All are defective

Steps:1. Number of ways to draw any 4 (10626)2. Number of ways to draw 4 defective (4C4)

D. At least 1 is defective

Steps: 1. Number of ways to draw any 4. (10626)2. Number of ways to draw no defective. (20C4)

Example 3

A store has 6 TV graphic magazines and 8 newstime magazines on the counter. If two customers purchased a magazine, find the probability that one of each magazine was purchased. Solution:

Example 4

A combination lock consists of the 26 letters of the alphabet. If a 3-letter combination is needed, find the probability that the combination will consist of the letter ABC in that order. The same letter can be used more than once. (Note: a combination lock is actually a permutation lock.)Solution:

Example 5

There are 8 married couples in a tennis club. If 1 man and 1 woman are selected at random to plan the summer tournament, find the probability that they are married to each other. Solution:

Exercises 4-51-17 odd

Wednesday