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Warm Up Find the 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

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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.

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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.

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Example 1: 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.

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Example 2: Try It Now 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 1st time does not affect the result of the 2nd roll. b. One student in your class is chosen for a project. Then another student in the class is chosen. Dependent; choosing the 1st student leaves fewer students to choose from the 2nd time.

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**Suppose an experiment involves flipping two fair coins**

Suppose an experiment involves flipping two fair coins. The sample space of outcomes is shown by the tree diagram. Determine the theoretical probability of both coins landing heads up.

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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.

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Example 3: 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.

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Example 3 (cont) P(red, green) = P(red) P(green) The probability of selecting red is , and the probability of selecting green is .

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Example 4: 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) • P(h) • P(h) The probability of landing heads up is with each event.

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**P(odd, odd) = P(odd) P(odd)**

Example 5: Try It Now 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) •

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**Suppose an experiment involves drawing marbles from a bag**

Suppose an experiment involves drawing marbles from a bag. Determine the 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

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**Suppose an experiment involves drawing marbles from a bag**

Suppose an experiment involves drawing marbles from a bag. Determine the 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

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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.

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Example 6: 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?

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**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 .

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Example 7: Try It Now 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?

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**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 .

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