Probability of Mutually Exclusive Events LESSON 13–6 Probability of Mutually Exclusive Events

Five-Minute Check (over Lesson 13–5) 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 Lesson Menu

Determine whether the event is independent or 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. A. independent B. dependent 5-Minute Check 1

Determine whether the event is independent or 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. A. independent B. dependent 5-Minute Check 2

A spinner with 4 congruent sectors labeled 1–4 is spun 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. 5-Minute Check 3

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. 5-Minute Check 4

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. Then/Now

mutually exclusive events complement Vocabulary

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. Example 1

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. Example 1

A. The events are 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. Example 1

A. The events are 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. A. The events are mutually exclusive. B. The events are not mutually exclusive. Example 1

Concept

Let Q represent picking a quarter. Let P represent picking a penny. 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. Example 2

P(Q or P) = P(Q) + P(P) Probability of mutually exclusive events Simplify. Answer: or about 41% ___ 9 22 Example 2

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. Example 2

Concept

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. Example 3

Let W represent watercolors and L represent landscapes. 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%. Example 3

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? A. B. C. D. Example 3

Concept

Probability of a complement 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. Example 4

Complementary Events Answer: The probability that one of Miguel’s tickets will not be selected is or about 93%. Example 4

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? A. B. C. D. Example 4

Concept

Identify and Use Probability Rules 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? Example 5

the first student chosen does not have a cat or a dog for a pet or Identify and Use Probability Rules Understand You 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. Example 5

Identify and Use Probability Rules Formulate The 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. Example 5

Probability of dependent events Identify and Use Probability Rules Determine Probability of dependent events Multiply. Example 5

Probability of complement 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%. Example 5

Identify and Use Probability Rules Justify Use 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. Evaluate In this example, it is more efficient to calculate the probability of the complement than to calculate the probability of each case in the set of desired outcomes. Example 5

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? A. 32% B. 39% C. 43% D. 56% Example 5

Probability of Mutually Exclusive Events LESSON 13–6 Probability of Mutually Exclusive Events