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40S Applied Math Mr. Knight – Killarney School Slide 1 Unit: Probability Lesson: PR-5 Independent and Dependent Events Independent and Dependent Events Learning Outcome B-4 PR-L5 Objectives: To identify independent and dependent events, and use the multiplication rule for probability.

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40S Applied Math Mr. Knight – Killarney School Slide 2 Unit: Probability Lesson: PR-5 Independent and Dependent Events In the average Manitoba high school, 40% of all the students get at least one cold each year. The public health nurse of 'Tween Lakes Collegiate encouraged all the students to take a certain brand of vitamin tablet to help reduce the number of colds. This year, 60% of the students did take the vitamin tablet, and 15% of the students who took the vitamin tablets also caught a cold. Did the vitamin treatment do any good? Was the probability of getting a cold reduced, increased, or not affected by the vitamins? In this lesson, we will learn how to answer this and other similar questions. Theory – Intro

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40S Applied Math Mr. Knight – Killarney School Slide 3 Unit: Probability Lesson: PR-5 Independent and Dependent Events Theory – Independent Events Event A and event B are independent events if the outcomes of event A in no way influence the outcomes of event B. The probability that all of a set of independent events will occur is the product of their separate probabilities. This may be written as the formula: (Note that the word 'and' implies multiplication.) For example, if you are flipping a coin and rolling a six-sided die, the outcome from rolling the die is in no way affected by the outcome of flipping the coin. Let A and B represent the following events: A. flipping a coin and getting heads B. rolling a die and getting 2 Then the following probabilities are:

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40S Applied Math Mr. Knight – Killarney School Slide 4 Unit: Probability Lesson: PR-5 Independent and Dependent Events Example – Independent Events

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40S Applied Math Mr. Knight – Killarney School Slide 5 Unit: Probability Lesson: PR-5 Independent and Dependent Events Example – Independent Events

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40S Applied Math Mr. Knight – Killarney School Slide 6 Unit: Probability Lesson: PR-5 Independent and Dependent Events Example cont’d – Independent Events

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40S Applied Math Mr. Knight – Killarney School Slide 7 Unit: Probability Lesson: PR-5 Independent and Dependent Events Example cont’d – Independent Events

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40S Applied Math Mr. Knight – Killarney School Slide 8 Unit: Probability Lesson: PR-5 Independent and Dependent Events We will take another look at the previous question: The probability of passing the English test is 2/3 and the probability of passing the science test is 3/4. What is the probability of passing at least one of the tests? Let E and S represent the following events: E: passing the English test S: passing the science test The probability of failing both tests is: Therefore, the probability of passing at least one test is the complement of failing both tests. We can write this as: Therefore, the probability of passing at least one test is 11/12 Theory – Complements

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40S Applied Math Mr. Knight – Killarney School Slide 9 Unit: Probability Lesson: PR-5 Independent and Dependent Events An "At Least Once" probability is the complement of the probability that the event will not occur. To determine an "at least once" probability, we: first calculate the probability that the event will not occur then find the complement of the probability that the event will not occur We can write this as a formula: P(the event occurs at least once) = 1 - P(the event does not occur) Theory – Complements & “At Least Once” Probabilities

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40S Applied Math Mr. Knight – Killarney School Slide 10 Unit: Probability Lesson: PR-5 Independent and Dependent Events Example – At Least Once

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40S Applied Math Mr. Knight – Killarney School Slide 11 Unit: Probability Lesson: PR-5 Independent and Dependent Events Event A and event B are dependent events if the outcomes of event A influence the outcomes of event B. The probability that all of a set of dependent events will occur is the product of their separate probabilities. This may be written as the formula: For example, if you have a bag containing six white and five black marbles, and randomly selected two marbles, what is the probability that both are black? The first marble is not returned to the bag before the second one is selected. Theory – Dependent Events

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40S Applied Math Mr. Knight – Killarney School Slide 12 Unit: Probability Lesson: PR-5 Independent and Dependent Events Let B1 and B2 represent the following events: B1: the first marble is black B2: the second marble is black Then, or stated another way, P(B2|B1) means "the probability of selecting a second black marble given that one black marble has been taken." After the first black marble has been taken, there are only four black marbles in the bag, and the total number of marbles in the bag is 10. For this reason: Theory – Dependent Events cont’d

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40S Applied Math Mr. Knight – Killarney School Slide 13 Unit: Probability Lesson: PR-5 Independent and Dependent Events After the first black marble has been taken, there are only four black marbles in the bag, and the total number of marbles in the bag is 10. For this reason: Theory – Dependent Events cont’d

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40S Applied Math Mr. Knight – Killarney School Slide 14 Unit: Probability Lesson: PR-5 Independent and Dependent Events A deck of cards consists of the following cards: Example – Dependent Events

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40S Applied Math Mr. Knight – Killarney School Slide 15 Unit: Probability Lesson: PR-5 Independent and Dependent Events Example – Dependent Events

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40S Applied Math Mr. Knight – Killarney School Slide 16 Unit: Probability Lesson: PR-5 Independent and Dependent Events Example – Dependent Events

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40S Applied Math Mr. Knight – Killarney School Slide 17 Unit: Probability Lesson: PR-5 Independent and Dependent Events Example – Dependent Events

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40S Applied Math Mr. Knight – Killarney School Slide 18 Unit: Probability Lesson: PR-5 Independent and Dependent Events The Moose have a game scheduled for December 30. If defense man Big A is able to play, the probability of winning is 0.8, but if he does not play, the probability of winning is 0.5. The probability that Big A will play is 0.3. Example – Dependent Events

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40S Applied Math Mr. Knight – Killarney School Slide 19 Unit: Probability Lesson: PR-5 Independent and Dependent Events The Moose have a game scheduled for December 30. If defense man Big A is able to play, the probability of winning is 0.8, but if he does not play, the probability of winning is 0.5. The probability that Big A will play is 0.3. Example cont’d – Dependent Events

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40S Applied Math Mr. Knight – Killarney School Slide 20 Unit: Probability Lesson: PR-5 Independent and Dependent Events There were 350 students in Math 101 at university. Of these 350 students, 60% had an 'A' in PreCalc 40S, and 30% failed Math 101. Statistics show that 10% of the students had an A in PreCalc 40S and still failed Math 101. Are getting an A in PreCalc and failing Math 101 independent events? Example – Independent or Dependent? Solution: Let A and B represent the following events: A: the event that a student got an A in PreCalc 40S B: the event that the student failed Math 101 Then P(A) and P(B) are: P(A) = 0.60 P(B) = 0.30, and P(A and B) = 0.60 x 0.30 = 0.18 This indicates that if getting an A in PreCalc and failing Math 101 were independent, then the number of students that got an A and failed should be 18% of the students. The statistics showed that only 10% of the students got an A in PreCalc and failed Math 101. Therefore, the two events are not independent. A student with an A in PreCalc is less likely to fail Math 101 at university (8% less likely).

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40S Applied Math Mr. Knight – Killarney School Slide 21 Unit: Probability Lesson: PR-5 Independent and Dependent Events Example – Independent or Dependent?

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