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Two cards are drawn from a deck of 52. Determine whether the events are independent or dependent. Find the indicated probability. A. selecting two face cards when the first card is replaced B. selecting two face cards when the first card is not replaced Warm up This unit • • • • • • • • • • • Fundamental counting principle factorial Permutations Combinations probability complement experimental probability theoretical probability independent events dependent events conditional probability Fundamental Counting Principle Lets start with a simple example. A student is to roll a die and flip a coin. How many possible outcomes will there be? Example 2A: Finding Permutations How many ways can a student government select a president, vice president, secretary, and treasurer from a group of 6 people? Example 3: Application There are 12 different-colored cubes in a bag. How many ways can Randall draw a set of 4 cubes from the bag? Check It Out! Example 1a A red number cube and a blue number cube are rolled. If all numbers are equally likely, what is the probability of the event? The sum is 6. Check It Out! Example 3 A DJ randomly selects 2 of 8 ads to play before her show. Two of the ads are by a local retailer. What is the probability that she will play both of the local retailer’s ads before her show? Check It Out! Example 5b The table shows the results of choosing one card from a deck of cards, recording the suit, and then replacing the card. Find the experimental probability of choosing a card that is not a club. 1. Find the probability of rolling a number greater than 2 and then rolling a multiple of 3 when a number cube is rolled twice. A drawer contains 8 blue socks, 8 black socks, and 4 white socks. Socks are picked at random. Explain why the events picking a blue sock and then another blue sock are dependent. Then find the probability. 7-5 Compound Events A simple event is an event that describes a single outcome. A compound event is an event made up of two or more simple events. Mutually exclusive events are events that cannot both occur in the same trial of an experiment. Rolling a 1 and rolling a 2 on the same roll of a number cube are mutually exclusive events. Holt McDougal Algebra 2 Remember! Recall that the union symbol means “or.” 7-5 Compound Events Example 1A: Finding Probabilities of Mutually Exclusive Events A group of students is donating blood during a blood drive. A student has a having type O blood and a probability of probability of having type A blood. Explain why the events “type O” and “type A” blood are mutually exclusive. Holt McDougal Algebra 2 7-5 Compound Events Example 1B: Finding Probabilities of Mutually Exclusive Events A group of students is donating blood during a blood drive. A student has a having type O blood and a probability of probability of having type A blood. What is the probability that a student has type O or type A blood? Holt McDougal Algebra 2 7-5 Compound Events Check It Out! Example 1a Each student cast one vote for senior class president. Of the students, 25% voted for Hunt, 20% for Kline, and 55% for Vila. A student from the senior class is selected at random. Explain why the events “voted for Hunt,” “voted for Kline,” and “voted for Vila” are mutually exclusive. Holt McDougal Algebra 2 7-5 Compound Events Check It Out! Example 1b Each student cast one vote for senior class president. Of the students, 25% voted for Hunt, 20% for Kline, and 55% for Vila. A student from the senior class is selected at random. What is the probability that a student voted for Kline or Vila? Holt McDougal Algebra 2 7-5 Compound Events Inclusive events are events that have one or more outcomes in common. When you roll a number cube, the outcomes “rolling an even number” and “rolling a prime number” are not mutually exclusive. The number 2 is both prime and even, so the events are inclusive. Holt McDougal Algebra 2 7-5 Compound Events There are 3 ways to roll an even number, {2, 4, 6}. There are 3 ways to roll a prime number, {2, 3, 5}. The outcome “2” is counted twice when outcomes are added (3 + 3) . The actual number of ways to roll an even number or a prime is 3 + 3 – 1 = 5. The concept of subtracting the outcomes that are counted twice leads to the following probability formula. Holt McDougal Algebra 2 7-5 Compound Events Remember! Recall that the intersection symbol means “and.” Holt McDougal Algebra 2 7-5 Compound Events Example 2A: Finding Probabilities of Compound Events Find the probability on a number cube. rolling a 4 or an even number Holt McDougal Algebra 2 7-5 Compound Events Example 2B: Finding Probabilities of Compound Events Find the probability on a number cube. rolling an odd number or a number greater than 2 Holt McDougal Algebra 2 7-5 Compound Events Check It Out! Example 2a A card is drawn from a deck of 52. Find the probability of each. drawing a king or a heart Holt McDougal Algebra 2 7-5 Compound Events Check It Out! Example 2b A card is drawn from a deck of 52. Find the probability of each. drawing a red card (hearts or diamonds) or a face card (jack, queen, or king) Holt McDougal Algebra 2 7-5 Compound Events Example 3: Application Of 1560 students surveyed, 840 were seniors and 630 read a daily paper. The rest of the students were juniors. Only 215 of the paper readers were juniors. What is the probability that a student was a senior or read a daily paper? Holt McDougal Algebra 2 7-5 Compound Events Example 3 Continued Step 1 Use a Venn diagram. Label as much information as you know. Being a senior and reading the paper are inclusive events. Holt McDougal Algebra 2 7-5 Compound Events Example 3 Continued Step 2 Find the number in the overlapping region. Step 3 Find the probability. Holt McDougal Algebra 2 7-5 Compound Events Check It Out! Example 3 Of 160 beauty spa customers, 96 had a hair styling and 61 had a manicure. There were 28 customers who had only a manicure. What is the probability that a customer had a hair styling or a manicure? Holt McDougal Algebra 2 7-5 Compound Events Example 4 Application Each of 6 students randomly chooses a butterfly from a list of 8 types. What is the probability that at least 2 students choose the same butterfly? P(at least 2 students choose same) = 1 – P(all choose different) Holt McDougal Algebra 2 7-5 Compound Events Example 4 Continued P(at least 2 students choose same) = 1 – 0.0769 ≈ 0.9231 The probability that at least 2 students choose the same butterfly is about 0.9231, or 92.31%. Holt McDougal Algebra 2 7-5 Compound Events Check It Out! Example 4 In one day, 5 different customers bought earrings from the same jewelry store. The store offers 62 different styles. Find the probability that at least 2 customers bought the same style. Holt McDougal Algebra 2 7-5 Compound Events Lesson Quiz: Part I You have a deck of 52 cards. 1. Explain why the events “choosing a club” and “choosing a heart” are mutually exclusive. 2. What is the probability of choosing a club or a heart? Holt McDougal Algebra 2 7-5 Compound Events Lesson Quiz: Part II The numbers 1–9 are written on cards and placed in a bag. Find each probability. 3. choosing a multiple of 3 or an even number 4. choosing a multiple of 4 or an even number 5. Of 570 people, 365 were male and 368 had brown hair. Of those with brown hair, 108 were female. What is the probability that a person was male or had brown hair? Holt McDougal Algebra 2 7-5 Compound Events Lesson Quiz: Part III 6. Each of 4 students randomly chooses a pen from 9 styles. What is the probability that at least 2 students choose the same style? Holt McDougal Algebra 2