Five-Minute Check (over Lesson 13–5)

CCSS

Then/Now

New Vocabulary

Example 1: Real-World Example: Identify Mutually Exclusive Events

Key Concept: Probability of Mutually Exclusive Events

Example 2: Real-World Example: Mutually Exclusive Events

Key Concept: Probability of Events That Are Not Mutually Exclusive

Example 3: Real-World Example: Events That Are Not Mutually Exclusive

Key Concept: Probability of the Complement of an Events

Example 4: Complementary Events

Concept Summary: Probability Rules

Example 5: Real-World Example: Identify and Use Probability Rules

Over Lesson 13–5

A. independent

B. dependent

Determine whether the event is independent or dependent. Samson ate a piece of fruit randomly from a basket that contained apples, bananas, and pears. Then Susan ate a second piece from the basket.

Over Lesson 13–5

A. independent

B. dependent

Determine whether the event is independent or dependent. Kimra received a passing score on the mathematics portion of her state graduation test. A week later, she received a passing score on the reading portion of the test.

Over Lesson 13–5

A spinner with 4 congruent sectors labeled 1–4 is spun. Then a die is rolled. What is the probability of getting even numbers on both events?

A. 1

B.

C.

D.

Over Lesson 13–5

Two representatives will be randomly chosen from a class of 20 students. What is the probability that Janet will be selected first and Erica will be selected second?

A.

B.

C.

D.

Content Standards

S.CP.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).

S.CP.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.

Mathematical Practices

1 Make sense of problems and persevere in solving them.

4 Model with mathematics.

You found probabilities of independent and dependent events.

• Find probabilities of events that are mutually exclusive and events that are not mutually exclusive.

• Find probabilities of complements.

• mutually exclusive events

• complement

Identify Mutually Exclusive Events

A. CARDS Han draws one card from a standard deck. Determine whether drawing an ace or a 9 is mutually exclusive or not mutually exclusive. Explain your reasoning.

Answer: These events are mutually exclusive. There are no common outcomes. A card cannot be both an ace and a 9.

Identify Mutually Exclusive Events

B. CARDS Han draws one card from a standard deck. Determine whether drawing a king or a club is mutually exclusive or not mutually exclusive. Explain your reasoning.

Answer: These events are not mutually exclusive. A king that is a club is an outcome that both events have in common.

A. The events are mutually exclusive.

B. The events are not mutually exclusive.

A. For a Halloween grab bag, Mrs. Roth has thrown in 10 caramel candy bars, 15 peanut butter candy bars, and 5 apples to have a healthy option. Determine whether drawing a candy bar or an apple is mutually exclusive or not mutually exclusive.

A. The events are mutually exclusive.

B. The events are not mutually exclusive.

B. For a Halloween grab bag, Mrs. Roth has thrown in 10 caramel candy bars, 15 peanut butter candy bars, and 5 apples to have a healthy option. Determine whether drawing a candy bar or something with caramel is mutually exclusive or not mutually exclusive.

Mutually Exclusive Events

COINS Trevor reaches into a can that contains 30 quarters, 25 dimes, 40 nickels, and 15 pennies. What is the probability that the first coin he picks is a quarter or a penny?

These events are mutually exclusive, since the coin picked cannot be both a quarter or a penny.

Let Q represent picking a quarter.

Let P represent picking a penny.

There are a total of 30 + 25 + 40 + 15 or 110 coins.

Mutually Exclusive Events

P(Q or P) = P(Q) + P(P) Probability ofmutually

exclusiveevents

Simplify.

Answer: or about 41%___922

MARBLES Hideki collects colored marbles so he can play with his friends. The local marble store has a grab bag that has 15 red marbles, 20 blue marbles, 3 yellow marbles and 5 mixed color marbles. If he reaches into a grab bag and selects a marble, what is the probability that he selects a red or a mixed color marble?

A. B.

C. D.

Events That Are Not Mutually Exclusive

ART Use the table below. What is the probability that Namiko selects a watercolor or a landscape?

Since some of Namiko’s paintings are both watercolors and landscapes, these events are not mutually exclusive. Use the rule for two events that are not mutually exclusive. The total number of paintings from which to choose is 30.

Events That Are Not Mutually Exclusive

Let W represent watercolors and L represent landscapes.

Substitution

Simplify.

Answer: The probability that Namiko selects

a watercolor or a landscape is or about 66%.

A. B.

C. D.

SPORTS Use the table. What is the probability that if a high school athlete is selected at random that the student will be a sophomore or a basketball player?

Complementary Events

GAMES Miguel bought 15 chances to pick the one red marble from a container to win a gift certificate to the bookstore. If there is a total of 200 marbles in the container, what is the probability Miguel will not win the gift certificate?

Let event A represent selecting one of Miguel’s tickets. Then find the probability of the complement of A.

Probability of a complement

Substitution

Subtract and simplify.

Complementary Events

Answer: The probability that one of Miguel’s tickets

will not be selected is or about 93%.

A. B.

C. D.

RAFFLE At a carnival, Sergio bought 18 raffle tickets, in order to win a gift certificate to the local electronics store. If there is a total of 150 raffle tickets sold, what is the probability Sergio will not win the gift certificate?

PETS A survey of Kingston High School students found that 63% of the students had a cat or a dog for a pet. If two students are chosen at random from a group of 100 students, what is the probability that at least one of them does not have a cat or a dog for a pet?

Identify and Use Probability Rules

UnderstandYou know that 63% of the students have a cat or a dog for a pet. The phrase at least one means one or more. So you need to find the probability that either

• the first student chosen does not have a cat or a dog for a pet or

• the second student chosen does not have a cat or a dog for a pet or

• both students chosen do not have a cat or a dog for a pet.

Identify and Use Probability Rules

PlanThe complement of the event described is that both students have a cat or a dog for a pet. Find the probability of this event, and then find the probability of its complement.

Let event A represent choosing a student that does have a cat or a dog for a pet.

Let event B represent choosing a student that does have a cat or a dog for a pet, after the first student has already been chosen.

These are two dependent events, since the outcome of the first event affects the probability of the outcome of the second event.

Identify and Use Probability Rules

Solve

Probability of dependent events

Multiply.

Identify and Use Probability Rules

Probability of complement

Substitution

Subtract.

Answer: So, the probability that at least one of the students does not have a cat or a dog for a

pet is or about 61%.

Identify and Use Probability Rules

Identify and Use Probability Rules

CheckUse logical reasoning to check the reasonableness of your answer.

The probability that one student chosen out of 100 does not have a cat or a dog for a pet is (100 – 63)% or 37%. The probability that two people chosen out of 100 have a cat or a dog for a pet should be greater than 37%. Since 61% > 37%, the answer is reasonable.

A. 32%

B. 39%

C. 43%

D. 56%

PETS A survey of Lakewood High School students found that 78% of the students preferred riding a bicycle to riding in a car. If two students are chosen at random from a group of 100 students, what is the probability that at least one of them does not prefer riding a bicycle to riding in a car?