4.4 Conditional Probability

Transcription

1 4.4 Conditional Probability It is often necessary to know the probability of an event under restricted conditions. Recall the results of a survey of 100 Grade 12 mathematics students in a local high school. Math Course Taken Advanced Functions and Introductory Calculus (AFIC) Number of Students Geometry and Discrete Math 33 Data Management 68 Geometry and Discrete Math and AFIC 30 Geometry and Discrete Math and Data Management Data Management and AFIC 0 All three courses The following example investigates a situation in which the probability of an event is determined under a restricted condition Example 1 Conditional Probability You are asked to determine the probability of selecting a Data Management student, but are first told that the only students from which you can choose are enrolled in AFIC. How does this additional condition affect the probability? Solution Without the condition that AFIC students are the only students left from which to select, the Venn diagram would look like the following: C 4 17 D S G 4.4 CONDITIONAL PROBABILITY 231

2 Based on the preceding diagram, the probability of selecting a Data 68 Management student from the entire group is P(D) When we add the condition that the students must be enrolled in AFIC, the Venn diagram changes. The sample space S does not consist of all possible students. It includes only those students taking AFIC. In the following Venn diagram, the new restricted sample space corresponds to the area shaded in blue. C 4 17 D S G The probability of selecting a Data Management student, given the condition that the student is also enrolled in AFIC, is called a conditional probability and is written as P(D C). The revised Venn diagram shows that the probability of D given that C has occurred is P(D C) n(c) We can use some simple algebra to convert this to a general statement about probabilities. P(D C) Divide the numerator and the denominator by n(s). For example, you can determine the probability of selecting a Data Management student as follows: P(D C) n(students enrolled in Data Management AND AFIC) n(students in AFIC) n(d C) n(d C) n(c) n(d C) n( S) n ( C) n( S) P(D C) P(C) P(D C) P(C) CHAPTER 4 DEALING WITH UNCERTAINTY AN INTRODUCTION TO PROBABILITY

3 Conditional Probability A A B U B The conditional probability of event B, given that event A has occurred, is given by P(B A) P(A B) P(A) Example 2 Conditional Probability in Weather Forecasting Suppose that in Vancouver the probability that a day will be both cloudy and rainy is 2%. Suppose further that 0% of all days are cloudy. Determine the probability that it will rain given it is a cloudy day in that city. Solution If C is the event that it is a cloudy day and R is the event that it is raining, the required probability is given by P(R C). This can be represented using the following Venn diagram that represents 100 days of Vancouver weather. Therefore, P(R C) P(R C) P(C) Cloudy 0 Rainy 2 Not Cloudy 0 There is a 0% chance of rain on a cloudy day. Rearranging the conditional probability formula results in another important relationship. Multiplication Law for Conditional Probability The probability of events A and B occurring, given that event A has occurred, is given by P(A B) P(B A) P(A) B) because P(B A) P(A P( A) multiply both sides by P(A) P(B A) P(A) P(A B) 4.4 CONDITIONAL PROBABILITY 233

4 Example 3 Multiplication Law for Conditional Probability What is the probability of drawing two aces in a row from a well-shuffled deck of 2 playing cards? The first card drawn is not replaced. Solution We want to know the probability that the first card is an ace and the second card is an ace. For the first card, there are 4 aces in the deck of 2 cards, so 4 P(first ace) 2 For the second card, given that the first is an ace, there are 3 aces remaining in the deck of 1 cards, so 3 P(second ace first ace) 1 Using the multiplication law, we get P(first ace and second ace) P(second ace first ace) P(first ace) KEY IDEAS conditional probability the conditional probability of event B, given that event A has occurred, is given by P(B A) P(A B) P(A) Multiplication Law for conditional probability the probability of events A and B occurring, given that event A has occurred, is given by P(A B) P(B A) P(A) 234 CHAPTER 4 DEALING WITH UNCERTAINTY AN INTRODUCTION TO PROBABILITY

5 4.4 Exercises A 1. Joel surveyed his class and summarized responses to the question, Do you like school? Liked Disliked No Opinion Total Males Females Total B Find each of the following probabilities. (a) P(likes school student is male) (b) P(student is female student dislikes school) 2. A person is chosen at random from shoppers at a department store. If the 2 person s probability of having blonde hair and glasses is 2 and the probability of wearing glasses is, 2 9 determine P(blonde hair wears glasses). 3. Tia and Jerry are tossing two coins. Tia wins when both coins turn up tails. The coins are tossed but roll under a chair. Jerry looks under the chair and, seeing both coins, says, At least one of them is tails. What is the probability that Tia wins? 4. What is the probability of being dealt two clubs in a row from a wellshuffled deck of 2 playing cards without replacing the first card drawn?. Knowledge and Understanding From a medical study of male patients, it was found that 200 were smokers; 720 died from lung cancer and of these, 610 were smokers. Determine (a) P(dying from lung cancer smoker) (b) P(dying from lung cancer non-smoker) 6. A bag contains three red marbles and five white marbles. What is the probability of drawing two red marbles at random if the first marble drawn is not replaced? 7. A road has two stop lights at two consecutive intersections. The probability of getting a green light at the first intersection is 0.6, and the probability of getting a green light at the second intersection, given that you got a green light at the first intersection, is 0.8. What is the probability of getting a green light at both intersections? 4.4 CONDITIONAL PROBABILITY 23

6 8. A survey of 1000 people asked whether they wear eyeglasses while driving. These people were also tested to see whether they need to wear eyeglasses while driving. The results are displayed in the table below. Wear Eyeglasses While Driving YES If a person is selected at random from this group, determine the probability he or she (a) should wear eyeglasses while driving (b) wears eyeglasses while driving (c) wears eyeglasses while driving even though he or she does not need to (d) does not wear eyeglasses while driving even though he or she needs to 9. Suppose the two joker cards are left in a standard deck of cards. One of the jokers is red and the other is black. A single card is drawn from the deck of 4 cards but not returned to the deck, and then a second card is drawn. Determine the probability of drawing (a) one of the jokers on the first draw and an ace on the second draw (b) a numbered card of any suit on the first draw and the red joker on the second draw (c) a queen on both draws (d) any black card on both draws (e) any numbered card below 10 on the first draw and the same number on a card on the second draw (f) the red joker or a red ace on either draw 10. Helena Maksimovik, the human resources director for a company, is given the task of hiring two salespeople from four candidates. From their résumés, the candidates could be ranked as follows: 1. Noel; 2. Sara; 3. Emil; 4. Fran. It is two days before Helena s scheduled vacation and she does not want to take the time to go through a formal interview process. Instead, she decides to hire two of the candidates at random. (a) List all the possible pairings that would make up the possible selections. (b) Determine the probability the selection will include (i) at least one of the top two candidates (ii) both of the top two candidates (iii) neither of the top two candidates (iv) Emil, if you know Sara has been selected (v) either Emil or Fran, if you know Sara has been selected NO Need to Wear YES Eyeglasses While Driving NO CHAPTER 4 DEALING WITH UNCERTAINTY AN INTRODUCTION TO PROBABILITY

7 C 11. A union and the management of a company are negotiating a new contract. History shows the following: Event A: Contract settlements are reached within two weeks 0% of the time. Event B: The union strike fund is large enough to support a strike 60% of the time. Events A and B: Both of the above conditions are satisfied 30% of the time. Determine the probability of each of the following. (a) A contract is settled in two weeks given that the strike fund is large enough to support a strike. (b) The union strike fund is large enough to support a strike given that a contract will be negotiated within two weeks. 12. Communication (a) With the aid of a Venn diagram, explain why the events in Question 11 are not mutually exclusive. (b) Explain why the answers to Question 11 suggest that event A and event B have no influence on one another. 13. Application Gwen has recently purchased a cottage. She has arranged to have a well dug on the property. One in five wells that were dug recently in the vicinity were dry, and 30% of the others are contaminated. Find the probability of each of the following. (a) Gwen s well will not be dry. (b) Gwen s well will be uncontaminated, given that it is not dry. (c) Gwen will have safe-drinking water from her well. (d) Gwen will not have safe-drinking water from her well. 14. A survey of readers of The News indicated that 40% of them also read The Chronicle, 32% read Info, and 11% read both publications. Find the probability that a reader of The News (a) also reads Info (b) reads Info, but not The Chronicle 1. There are three cards in a hat. One card is black on both sides and the other two cards are black on one side and white on the other. If a card is drawn randomly from the hat and placed on a table so that the underside is not visible, determine the probability that the back of the card is black if the front is showing black. 4.4 CONDITIONAL PROBABILITY 237

8 16. Thinking, Inquiry, Problem Solving The quality-control inspector for a computer company either accepts or rejects shipments of microprocessors as a result of testing a sample of the items in a shipment. The inspector s previous performance indicates that she has accepted 98% and rejected 2% of all shipments that turned out to be good accepted 94% of all shipments even though % of the shipments are known to be inferior (a) Find the probability that a good shipment is rejected. (b) Find the probability that an inferior shipment is accepted. ADDITIONAL ACHIEVEMENT CHART QUESTIONS 17. Knowledge and Understanding A die is rolled twice. Determine the probability that the sum of the two rolls is greater than 6, given that the first roll is a Communication A poll was taken to see how people felt about a plan to build a new community centre in town. Use the results to create a conditional probability question. In Favour Opposed Total Retired Non-Retired Student Total Application A jar contains six red marbles and four green ones. If two marbles are drawn at random from the jar, and the first marble is not returned to the jar, find the probability of each of these events. (a) the second marble is green, given that the first is red (b) both marbles are red (c) both marbles are green (d) the second marble is red 20. Thinking, Inquiry, Problem Solving Prove that P(A B) P(B A). P(A) P(B) 238 CHAPTER 4 DEALING WITH UNCERTAINTY AN INTRODUCTION TO PROBABILITY