Warm Up Find the theoretical probability of each outcome 1. rolling a 6 on a number cube.

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

Learning Targets Find the probability of independent events. Find the probability of 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.

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: Identifying situations involving independent and dependent events 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.

On Your Own! 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. b. One student in your class is chosen for a project. Then another student in the class is chosen.

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.

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.

The probability of selecting red is , and the probability of selecting green is . Example 2: 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. P(red, green) = P(red)  P(green)

The probability of landing heads up is with each event. 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)

On Your Own Example 2 An experiment consists of spinning the spinner twice. What is the probability of spinning two odd numbers? .

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 Probability of drawing a red marble on the first draw P(Drawing 2 red Marbles) = 1/3 * ¼ = 1/12

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?

1 • 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. Understand the Problem Example 3 Continued 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:

Make a Plan Iris chooses from: Grant chooses from: pretzels chips 2 Example 3 Continued Draw a diagram. After Grant selects a bag, the sample space changes. So the events are dependent. After Grant selects a bag, the sample space changes. So the events are dependent.

3 Solve P(pretzel and chip) = P(pretzel) P(chip after pretzel) • The probability that Grant selects a bag of pretzels and Iris selects a bag of chips is . Example 3 Continued 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.

Look Back 4 Example 3 Continued 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.

On your Own! Application 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?

Definition of Odds: A ratio expressing the likelihood of an event. Assume all outcomes are equally likely, and that there are m favorable an n unfavorable outcomes. The odds for the event -> m:n The odds against the event -> n:m

Example 4 – Calculating Odds • A bag contains 6 red marbles, 2 yellow marbles, and 1 blue marble. • What are the odds of drawing a red marble? • * Favorable: unfavorable • 6 red marbles: 3 non red marbles • 6:3 or 2:1 • b) What are the odds against drawing a blue marble? • * unfavorable: favorable • 8 non blue marbles: 1 blue marble • 8:1

On your own – Example 4 • A box contains 3 pink, 4 yellow, and 5 blue highlighters. One highlighter is chosen at random. • What are the odds of choosing a yellow highlighter? • What are the odds against choosing a blue highlighter?

Warm Up Find the theoretical probability of each outcome 1. rolling a 6 on a number cube.