Independent and Dependent Events 10 7 Dependent Events

Independent and Dependent Events 10 -7 Dependent Events Warm Up Lesson Presentation Lesson Quiz Holt Mc. Dougal Algebra 1 Algebra 11 Holt Mc. Dougal

10 -7 Independent and Dependent Events Warm Up Find theoretical probability of each outcome 1. rolling a 6 on a number cube. 2. rolling an odd number on a number cube. 3. flipping two coins and both landing head up Holt Mc. Dougal Algebra 1

10 -7 Independent and Dependent Events Objectives Find the probability of independent events. Find the probability of dependent events. Holt Mc. Dougal Algebra 1

10 -7 Independent and Dependent Events Vocabulary independent events Holt Mc. Dougal Algebra 1

10 -7 Independent and Dependent Events Adam’s teacher gives the class two list of titles and asks each student to choose two of them to read. Adam can choose one title from each list or two titles from the same list. Holt Mc. Dougal Algebra 1

10 -7 Independent and Dependent Events are independent events if the occurrence of one event does not affect the probability of the other. Events are dependent events if the occurrence of one event does affect the probability of the other. Holt Mc. Dougal Algebra 1

10 -7 Independent and Dependent Events Example 1: Classifying Events as Independent or Dependent Tell whether each set of events is independent or dependent. Explain you answer. A. You select a card from a standard deck of cards and hold it. A friend selects another card from the same deck. Dependent; your friend cannot pick the card you picked and has fewer cards to choose from. B. You flip a coin and it lands heads up. You flip the same coin and it lands heads up again. Independent; the result of the first toss does not affect the sample space for the second toss. Holt Mc. Dougal Algebra 1

10 -7 Independent and Dependent Events Check It Out! Example 1 Tell whether each set of events is independent or dependent. Explain you answer. a. A number cube lands showing an odd number. It is rolled a second time and lands showing a 6. Independent; the result of rolling the number cube the 1 st time does not affect the result of the 2 nd roll. b. One student in your class is chosen for a project. Then another student in the class is chosen. Dependent; choosing the 1 st student leaves fewer students to choose from the 2 nd time. Holt Mc. Dougal Algebra 1

10 -7 Independent and Dependent Events Suppose an experiment involves flipping two fair coins. The sample space of outcomes is shown by the tree diagram. Determine theoretical probability of both coins landing heads up. Holt Mc. Dougal Algebra 1

10 -7 Independent and Dependent Events Now look back at the separate theoretical probabilities of each coin landing heads up. The theoretical probability in each case is. The product of these two probabilities is , the same probability shown by the tree diagram. To determine the probability of two independent events, multiply the probabilities of the two events. Holt Mc. Dougal Algebra 1

10 -7 Independent and Dependent Events Holt Mc. Dougal Algebra 1

10 -7 Independent and Dependent Events Example 2 A: Finding the Probability of Independent Events An experiment consists of randomly selecting a marble from a bag, replacing it, and then selecting another marble. The bag contains 3 red marbles and 12 green marbles. What is the probability of selecting a red marble and then a green marble? Because the first marble is replaced after it is selected, the sample space for each selection is the same. The events are independent. Holt Mc. Dougal Algebra 1

10 -7 Independent and Dependent Events Example 2 A Continued P(red, green) = P(red) P(green) The probability of selecting red is , and the probability of selecting green is Holt Mc. Dougal Algebra 1 .

10 -7 Independent and Dependent Events Example 2 B: Finding the Probability of Independent Events A coin is flipped 4 times. What is the probability of flipping 4 heads in a row. Because each flip of the coin has an equal probability of landing heads up, or a tails, the sample space for each flip is the same. The events are independent. P(h, h, h, h) = P(h) • P(h) The probability of landing heads up is with each event. Holt Mc. Dougal Algebra 1

10 -7 Independent and Dependent Events Check It Out! Example 2 An experiment consists of spinning the spinner twice. What is the probability of spinning two odd numbers? The result of one spin does not affect any following spins. The events are independent. With 6 numbers on the spinner, 3 of which are odd, the probability of landing on two odd numbers is. P(odd, odd) = P(odd) • P(odd) Holt Mc. Dougal Algebra 1

10 -7 Independent and Dependent Events Suppose an experiment involves drawing marbles from a bag. Determine theoretical probability of drawing a red marble and then drawing a second red marble without replacing the first one. Probability of drawing a red marble on the first draw Holt Mc. Dougal Algebra 1

10 -7 Independent and Dependent Events Suppose an experiment involves drawing marbles from a bag. Determine theoretical probability of drawing a red marble and then drawing a second red marble without replacing the first one. Probability of drawing a red marble on the second draw Holt Mc. Dougal Algebra 1

10 -7 Independent and Dependent Events To determine the probability of two dependent events, multiply the probability of the first event times the probability of the second event after the first event has occurred. Holt Mc. Dougal Algebra 1

10 -7 Independent and Dependent Events Example 3: Application A snack cart has 6 bags of pretzels and 10 bags of chips. Grant selects a bag at random, and then Iris selects a bag at random. What is the probability that Grant will select a bag of pretzels and Iris will select a bag of chips? Holt Mc. Dougal Algebra 1

10 -7 Independent and Dependent Events Example 3 Continued 1 Understand the Problem The answer will be the probability that a bag of chips will be chosen after a bag of pretzels is chosen. List the important information: • Grant chooses a bag of pretzels from 6 bags of pretzels and 10 bags of chips. • Iris chooses a bag of chips from 5 bags of pretzels and 10 bags of chips. Holt Mc. Dougal Algebra 1

10 -7 Independent and Dependent Events Example 3 Continued 2 Make a Plan Draw a diagram. After Grant selects a bag, the sample space changes. So the events are dependent. Iris chooses from: Grant chooses from: pretzels chips After Grant selects a bag, the sample space changes. So the events are dependent. Holt Mc. Dougal Algebra 1

10 -7 Independent and Dependent Events Example 3 Continued 3 Solve P(pretzel and chip) = P(pretzel) • P(chip after pretzel) Grant selects one of 6 bags of pretzels from 16 total bags. Then Iris selects one of 10 bags of chips from 15 total bags. The probability that Grant selects a bag of pretzels and Iris selects a bag of chips is. Holt Mc. Dougal Algebra 1

10 -7 Independent and Dependent Events Example 3 Continued 4 Look Back Drawing a diagram helps you see how the sample space changes. This means the events are dependent, so you can use the formula for probability of dependent events. Holt Mc. Dougal Algebra 1

10 -7 Independent and Dependent Events Check It Out! Example 3 A bag has 10 red marbles, 12 white marbles, and 8 blue marbles. Two marbles are randomly drawn from the bag. What is the probability of drawing a blue marble and then a red marble? Holt Mc. Dougal Algebra 1

10 -7 Independent and Dependent Events Check It Out! Example 3 Continued 1 Understand the Problem The answer will be the probability that a red marble will be chosen after a blue marble is chosen. List the important information: • A blue marble is chosen from a bag containing 10 red, 12 white, and 8 blue marbles. • Then a red marble is chosen from a bag of 10 red, 12 white, and 7 blue marbles. Holt Mc. Dougal Algebra 1

10 -7 Independent and Dependent Events Check It Out! Example 3 Continued 2 Make a Plan Draw a diagram. First choice from: Second choice from: After the first selection, the sample space changes. So the events are dependent. Holt Mc. Dougal Algebra 1

10 -7 Independent and Dependent Events Check It Out! Example 3 Continued 3 Solve P(blue and red) = P(blue) • P(red after blue) One of 8 blue marbles is selected from a total of 30 marbles. Then one of 10 red marbles is selected from the 29 remaining marbles. The probability that first a blue marble is selected and then a red marble is selected is Holt Mc. Dougal Algebra 1 .

10 -7 Independent and Dependent Events Check It Out! Example 3 Continued 4 Look Back Drawing a diagram helps you see how the sample space changes. This means the events are dependent, so you can use the formula for probability of dependent events. Holt Mc. Dougal Algebra 1

10 -7 Independent and Dependent Events Lesson Quiz: Part I Tell whether each set of events is independent or dependent. Explain your answer. 1. flipping two different coins and each coin landing showing heads Independent; the flip of the first coin does not affect the sample space for the flip of the second coin. 2. drawing a red card from a standard deck of cards and not replacing it; then drawing a black card from the same deck of cards Dependent; there are fewer cards to choose from when drawing the black card. Holt Mc. Dougal Algebra 1

10 -7 Independent and Dependent Events Lesson Quiz: Part II 3. Eight cards are numbered from 1 to 8 and placed in a box. One card is selected at random and not replaced. Another card is randomly selected. What is the probability that both cards are greater than 5? 4. An experiment consists of randomly selecting a marble from a bag, replacing it, and then selecting another marble. The bag contains 3 yellow marbles and 2 white marbles. What is the probability of selecting a white marble and then a yellow marble? Holt Mc. Dougal Algebra 1

10 -7 Independent and Dependent Events Lesson Quiz: Part III 5. A number cube is rolled two times. What is the probability of rolling an even number first and then a number less than 3? Holt Mc. Dougal Algebra 1