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U NIT : P ROBABILITY 9-7: P ROBABILITY OF M ULTIPLE E VENTS Essential Question: How do you determine if two events, A and B, are independent or dependent, and whether they are mutually exclusive?

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9-7: P ROBABILITY OF M ULTIPLE E VENTS Dependent Event: When the outcome of one event affects the outcome of a second event. Examples: Pick a flower from a garden. Then pick another flower from the same garden. Independent Event: When the outcome of one event does not affect the outcome of a second event. Example: Roll a number cube. Then toss a coin. Suppose you select a marble from a bag of marbles. You replace the marble and then select again. Are your selections dependent or independent? Explain. Independent; the number of marbles is the same after the marble is replaced.

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9-7: P ROBABILITY OF M ULTIPLE E VENTS If A and B are independent events, then P(A and B) = P(A) P(B) Example: Suppose your favorite radio station is running a promotional campaign. Every hour, four callers at random get to select two songs each. You call the station once after 7:00 A.M. and again after 3:00 P.M. What is the probability that you will be one of the four callers both times you call? P(A and B) = P(A) P(B) = 4 / 125 4 / 200 = 16 / 25,000 = 2 / 3125 The probability is 2 / 3125, or 0.064% Suppose the radio station chooses 5 callers at random. What is the probability of being picked each time you call? HourCalls Received 7:00 A. M. 3:00 P. M. 125 200 5 / 125 5 / 200 = 25 / 25,000 = 1 / 1000, or about 0.1%

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9-7: P ROBABILITY OF M ULTIPLE E VENTS When two events cannot happen at the same time, we say they are mutually exclusive. Mutually exclusive: Rolling a 2 or a 3 on a number cube. Not mutually exclusive: Rolling an even number or a multiple of 3 on a number cube. 6 is both an even number and a multiple of 3. Are the events mutually exclusive? Explain. Rolling an even number and rolling a prime number on a number cube. Rolling an even number and rolling a number less than 2 on a number cube. Not mutually exclusive (2 is both even and prime) Mutually exclusive since no number less than 2 is even

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9-7: P ROBABILITY OF M ULTIPLE E VENTS If A and B are mutually exclusive events, then P(A or B) = P(A) + P(B) Example: About 53% of US college students are under 25 years old. About 21% of US college students are over 34 years old. What is the probability that a US college student chosen at random is under 25 or over 34? Since a student can’t be under 25 and over 34 at the same time, these events are mutually exclusive, so P(under 25 or over 34)= P(under 25) + P(over 34) = 0.53 + 0.21 = 0.74 What is the probability of a student being between 25 – 34 or over 34? 0.47, or 47%

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9-7: P ROBABILITY OF M ULTIPLE E VENTS If A and B are not mutually exclusive events, then P(A or B) = P(A) + P(B) – P(A and B) Suppose you have a bowl of fruit with the following: 3 red apples, 2 green apples, 2 orange oranges, 1 green lime, 1 yellow lemon Example: Selecting a piece of fruit at random, what is the probability that the fruit is an apple or green? Because there are pieces that satisfy both conditions (green apples), the events are not mutually exclusive, so P(green or apple)= P(green) + P(apple) – P(green and apple) = 3 / 9 + 5 / 9 – 2 / 9 = 6 / 9 = 2 / 3, or 67% What is the probability that the fruit is an apple or red? 5 / 9, or 56%

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9-7: P ROBABILITY OF M ULTIPLE E VENTS Assignment Page 534 – 535 Problems 1 – 25, odd

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