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**Probability of Multiple Events (Independent and Dependent Events)**

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Warm Up (For help, go to Lesson 1-6.) A bag contains 24 green marbles, 22 blue marbles, 14 yellow marbles, and 12 red marbles. Suppose you pick one marble at random. Find each probability. 1. P(yellow) 2. P(not blue) 3. P(green or red)

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Warm Up - Solutions 1. total number of marbles = = 72 P(yellow) = = = 2. total number of marbles = = 72 P(not blue) = = = = 3. total number of marbles = = 72 P(green or red) = = = = 14 72 72 – 22 2 • 7 2 • 36 7 36 50 2 • 25 25 1 • 36 1 2 /

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**Warm Up Of 300 senior students at Howe High, 150 have taken physics,**

192 have taken chemistry, and 30 have taken neither physics nor chemistry. How many students have taken both physics and chemistry?

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Warm Up - Solutions Let x = the number of students who have taken both physics and chemistry. Then (150 – x) is the number of students who have taken physics, but not chemistry. And (192 – x) is the number of students who have taken chemistry, but not physics. 30 students have taken neither, and there are 300 students altogether. x + (150 – x) + (192 – x) + 30 = 300 ( ) + (1 – 1 – 1)x = 300 372 – x = 300 372 – 300 = x 72 = x So, 72 students have taken both physics and chemistry.

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**Consider the Following:**

A marble is picked at random from a bag. Without putting the marble back, a second one has chosen. How does this affect the probability? A card is picked at random from a deck of cards. Then a dice is rolled. How does this affect the probability?

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**Outcomes of Different Events**

When the outcome of one event affects the outcome of a second event, we say that the events are dependent. When one outcome of one event does not affect a second event, we say that the events are independent.

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**Probability of Multiple Events**

Classify each pair of events as dependent or independent. a. Spin a spinner. Select a marble from a bag that contains marbles of different colors. Since the two events do not affect each other, they are independent. b. Select a marble from a bag that contains marbles of two colors. Put the marble aside, and select a second marble from the bag. Picking the first marble affects the possible outcome of picking the second marble. So the events are dependent.

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**Decide if the following are dependent or independent**

An expo marker is picked at random from a box and then replaced. A second marker is then grabbed at random. Two dice are rolled at the same time. An Ace is picked from a deck of cards. Without replacing it, a Jack is picked from the deck. Independent Independent Dependent

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**How to find the Probability of Two Independent Events**

If A and B are independent events, the P(A and B) = P(A) * P(B) Ex: If P(A) = ½ and P(B) = 1/3 then P(A and B) =

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Let’s Try One A box contains 20 red marbles and 30 blue marbles. A second box contains 10 white marbles and 47 black marbles. If you choose one marble from each box without looking, what is the probability that you get a blue marble and a black marble? 30 50 47 57 Relate: probability of both events is probability of first event times probability of second event Define: Event A = first marble is blue. Then P(A) = . Event B = second marble is black. Then P(B) = . Write: P(A and B) = P(A) • P(B) P(A and B) = • 30 50 47 57 1410 2850 = . = Simplify. 47 95 The probability that a blue and a black marble will be drawn is , or 49%. 47 95

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**Mutually Exclusive Events**

Two events are mutually exclusive then they can not happen at the same time.

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**Probability of Multiple Events**

Are the events mutually exclusive? Explain. a. rolling an even number or a prime number on a number cube By rolling a 2, you can roll an even number and a prime number at the same time. So the events are not mutually exclusive. b. rolling a prime number or a multiple of 6 on a number cube Since 6 is the only multiple of 6 you can roll at a time and it is not a prime number, the events are mutually exclusive.

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**How to find the Probability of Two Mutually Exclusive Events**

If A and B are mutually exclusive events, then P(A or B) = P(A) + P(B) If A and B are not mutually exclusive events, then P(A or B) = P(A) + P(B) – P(A B)

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Let’s Try Some A spinner has ten equal-sized sections labeled 1 to 10. Find the probability of each event. No! P(A)+P(B)-P(A B) P(even or multiple 0f 5) P(Multiple of 3 or 4) Yes! P(A)+P(B) Hint: Decide if each event is mutually exclusive

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Let’s Try Some P(even or multiple 0f 5) P(Multiple of 3 or 4)

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**Let’s Try One Solution: .33 + .28 = .61 = 61%**

At a restaurant, customers get to choose one of four desserts. About 33% of the customers choose Crème Brule, and about 28% Chocolate Cheese Cake. Kayla is treating herself for pole vaulting four feet at the meet. What is the probability that Kayla will choose Crème Brule or Chocolate Cheese Cake? Yes. So: P(A) + P(B) Are the events mutually exclusive? Solution: = .61 = 61%

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**Probability of Multiple Events**

A spinner has twenty equal-size sections numbered from 1 to 20. If you spin the spinner, what is the probability that the number you spin will be a multiple of 2 or a multiple of 3? No. So: P(A) + P(B) - P(AB) Are the events mutually exclusive? P(multiple of 2 or 3) = P (multiple of 2) + P (multiple of 3) – P (multiple of 2 and 3) = + – 10 20 6 3 = 13 20 The probability of spinning a multiple of 2 or 3 is . 13 20

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