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Probability of Independent and Dependent Events and Review Probability & Statistics 1.0 Students know the definition of the notion of independent events and can use the rules for addition, multiplication, and complementation to solve for probabilities of particular events in finite sample spaces. 2.0 Students know the definition of conditional probability and use it to solve for probabilities in finite sample spaces.

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Probability of Independent and Dependent Events and Review Objectives Solve for the probability of an independent event. Solve for the probability of a dependent event. Key Words Independent Events – The occurrence of one event does not affect the occurrence of the other Dependents Events – The occurrence of one event does affect the occurrence of the other Conditional Probability – Two dependent events A and B, the probability that B will occur given that A has occurred.

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Example 1 Identify Events Tell whether the events are independent or dependent. Explain. a.Your teacher chooses students at random to present their projects. She chooses you first, and then chooses Kim from the remaining students. b.You flip a coin, and it shows heads. You flip the coin again, and it shows tails. c.One out of 25 of a model of digital camera has some random defect. You and a friend each buy one of the cameras. You each receive a defective camera.

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Example 1 Identify Events SOLUTION a.Dependent; after you are chosen, there is one fewer student from which to make the second choice. b.Independent; what happens on the first flip has no effect on the second flip. c.Independent; because the defects are random, whether one of you receives a defective camera has no effect on whether the other person does too.

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Checkpoint Identify Events ANSWER dependent You choose Alberto to be your lab partner. Then Tia chooses Shelby. 1. Tell whether the events are independent or dependent. Explain. You spin a spinner for a board game, and then you roll a die. 2. ANSWER independent

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Example 2 Find Conditional Probabilities Concerts A high school has a total of 850 students. The table shows the numbers of students by grade at the school who attended a concert. a.What is the probability that a student at the school attended the concert? b.What is the probability that a junior did not attend the concert? Freshman Sophomore Junior Senior 80 132 179 173 GradeAttended Did not attend 120 86 51 29

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Example 2 Find Conditional Probabilities 564 850 = ~ ~ 0.664 SOLUTION a. 80 P (attended) total who attended total students = 850 = +++ 132173179 b. P (did not attend junior) = juniors who did not attend total juniors = 29 173 + 29 202 = 0.144 ~ ~ 29

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Checkpoint Use the table below to find the probability that a student is a junior given that the student did not attend the concert. 3. ANSWER 29 286 0.101 ~ ~ Find Conditional Probabilities Freshman Sophomore Junior Senior 80 132 179 173 GradeAttended Did not attend 120 86 51 29

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Probability of Independent and Dependent Events Independent Events – If A and B are independent events, then the probability that both A and B occur is P(A and B)=P(A)*P(B) Dependent Events – If A and B are dependent events, then the probability that both A and B occur is P(A and B)=P(A)*P(B|A)

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Example 3 Independent and Dependent Events Games A word game has 100 tiles, 98 of which are letters and two of which are blank. The numbers of tiles of each letter are shown in the diagram. Suppose you draw two tiles. Find the probability that both tiles are vowels in the situation described. a.You replace the first tile before drawing the second tile. b.You do not replace the first tile before drawing the second tile.

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Example 3 Independent and Dependent Events SOLUTION a.If you replace the first tile before selecting the second, the events are independent. Let A represent the first tile being a vowel and B represent the second tile being a vowel. Of 100 tiles, ++++= 91298442 are vowels. ( ( P = A ( ( PB ( ( P = 42 100 = 0.1764 A and B 42 100

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Example 3 Independent and Dependent Events b.If you do not replace the first tile before selecting the second, the events are dependent. After removing the first vowel, 41 vowels remain out of 99 tiles. = A ( ( PB ( ( P = 42 100 41 99 0.1739A ~ ~ | ( ( P A and B

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Checkpoint Find Probabilities of Independent and Dependent Events In the game in Example 3, you draw two tiles. What is the probability that you draw a Q, then draw a Z if you first replace the Q ? What is the probability that you draw both of the blank tiles (without replacement)? 4. ANSWER 1 10,000 = 0.0001 ; 1 4950 ~ ~ 0.0002

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Conclusion Summary How are probabilities calculated for two events when the outcome of the first event influences the outcome of the second event? – Multiply the probability of the second event, given that the first event happen. Assignment Probability of Independent and Dependent Events – Page 572 – #(11-14,15,18,22,26,30)

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Review Probability & Statistics 1.0 Students know the definition of the notion of independent events and can use the rules for addition, multiplication, and complementation to solve for probabilities of particular events in finite sample spaces. 2.0 Students know the definition of conditional probability and use it to solve for probabilities in finite sample spaces.

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Theoretical Probability of an Event Example: What is the probability that the spinner shown lands on red if it is equally likely to land on any section? Solution: The 8 sections represent the 8 possible outcomes. Three outcomes correspond to the event lands on red. P(red) = 3 8 Number of outcomes in event Total number of outcomes =

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Experimental Probability of an Event Example: Surveys The graph shows results of a survey asking students to name their favorite type of footwear. What is the experimental probability that a randomly chosen student prefers (a)Sneakers? (b)Shoes or boots? P (prefers sneakers) Number preferring sneakers Total number of students = 820 1700 = ~ ~ 0.48 P (prefers shoes or boots) Number preferring shoes or boots Total number of students = = 340 1700 ~ ~ 0.19

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Probability of Compound Events

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Probability of the Complement of an Event Recall: Complement of an Event All outcomes that are not in the event

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Probability of Independent and Dependent Events Independent Events – If A and B are independent events, then the probability that both A and B occur is P(A and B)=P(A)*P(B) Dependent Events – If A and B are dependent events, then the probability that both A and B occur is P(A and B)=P(A)*P(B|A)

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