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For independent events, the occurrence of one event has no effect on the probability that a second event will occur. For dependent events, the occurrence of one event does have an effect on the probability that a second event will occur.

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**Decide whether the set of events are dependent or independent**

Decide whether the set of events are dependent or independent. Explain your answer. Kathi draws a 4 from a set of cards numbered 1–10 and rolls a 2 on a number cube.

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**Decide whether the set of events are dependent or independent**

Decide whether the set of events are dependent or independent. Explain your answer. Yuki chooses a book from the shelf to read, and then Janette chooses a book from the books that remain.

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**Decide whether each set of events is independent or dependent**

Decide whether each set of events is independent or dependent. Explain your answer. Tara randomly draws a card from a deck of playing cards, looks at it, and puts it back in the deck. Jerry randomly draws a card from the same deck.

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**There are five prizes hidden in boxes labeled 1 through 5**

There are five prizes hidden in boxes labeled 1 through 5. Nelson chooses one, and then Melanie chooses one of the remaining four boxes.

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**P(A and B) P(A) P(B) = Probability of Two Independent Events**

To find the probability that two independent events will happen, multiply the probabilities of the two events. Probability of Two Independent Events P(A and B) = P(A) P(B) • Probability of both events Probability of first event Probability of second event

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Find the probability of choosing a green marble at random from a bag containing 5 green and 10 white marbles and then flipping a coin and getting tails. The outcome of choosing the marble does not affect the outcome of flipping the coin, so the events are independent. P(green and tails) = P(green) · P(tails)

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You draw a marble from a bag containing 3 red and 7 white marbles and then flip a coin. Find the probability for each pair of events. drawing a red marble and getting heads The outcome of choosing a marble does not affect the coin toss, so the events are independent. P(red marble and heads) = P(red marble) P(heads)

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You draw a marble from a bag containing 3 red and 7 white marbles and then flip a coin. Find the probability for each pair of events. drawing a white marble and getting tails The events are independent. P (white marble and tails) = P (white marble) P (tails)

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**P(A and B) P(A) P(B after A) = Probability of Two Dependent Events**

To find the probability of two dependent events, you must determine the effect that the first event has on the probability of the second event. Probability of Two Dependent Events P(A and B) = P(A) P(B after A) • Probability of both events Probability of first event Probability of second event

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**Additional Example 3: Finding the Probability of Dependent Events**

A reading list contains 5 historical books and 3 science-fiction books. What is the probability that Juan will randomly choose a historical book for his first report and a science-fiction book for his second? The first choice changes the number of books left, and may change the number of science-fiction books left, so the events are dependent.

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**Additional Example 3 Continued**

P(historical and then science-fiction) = P(A) · P(B after A) 15 56

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Check It Out: Example 3 Juan’s mp3 playlist has 7 dance tracks and 3 rock tracks. What is the probability that his player randomly selects a dance track followed by a rock track?

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Lesson Quiz: Part I Decide whether each event is independent or dependent. Explain. 1. Mary chooses a game piece from a board game, and then Jason chooses a game piece from three remaining pieces. 2. Find the probability of spinning an evenly divided spinner numbered 1–8 and getting a composite number on one spin and getting an odd number on a second spin.

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Lesson Quiz: Part II 3. Sarah picks 2 hats at random from 5 bill caps and 3 beanies. What is the probability that both are bill caps?

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**Lesson Quiz for Student Response Systems**

1. Decide whether the given event is independent or dependent, and then explain. Regina flips tails on a coin and rolls 5 on a number cube. A. independent; the outcome of tossing a coin does not affect the out come of rolling a number cube. B. independent; the outcome of tossing a coin affects the out come of rolling a number cube. C. dependent; the outcome of tossing a coin does not affect the out come of rolling a number cube. D. dependent; the outcome of tossing a coin affects the out come of rolling a number cube. 16 16 16 16

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**Lesson Quiz for Student Response Systems**

2. What is the probability that Lupe will choose 2 apples from a bin of 10 apples, 8 bananas, and 7 oranges? A. B. C. D. 17 17 17 17

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