Independent and Dependent Events

1. Independent and Dependent Events PR-L5 Objectives:To identify independent and dependent events, and use the multiplication rule for probability. Learning Outcome B-4

2. 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

3. 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: • flipping a coin and getting heads • rolling a die and getting 2 • Then the following probabilities are: Theory – Independent Events

4. Example – Independent Events

5. Example – Independent Events

6. Example cont’d – Independent Events

7. Example cont’d – Independent Events

8. 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 testS: passing the science testThe 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

9. 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

10. Example – At Least Once

11. 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

12. Let B1 and B2 represent the following events:B1: the first marble is blackB2: the second marble is blackThen, 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

13. 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

14. A deck of cards consists of the following cards: Example – Dependent Events

15. Example – Dependent Events

16. Example – Dependent Events

17. Example – Dependent Events

18. 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

19. 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

20. 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? Solution:Let A and B represent the following events:A: the event that a student got an A in PreCalc 40SB: the event that the student failed Math 101Then P(A) and P(B) are:P(A) = 0.60P(B) = 0.30, andP(A and B) = 0.60 x 0.30 = 0.18This 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). Example – Independent or Dependent?

21. Example – Independent or Dependent?