156 CHAPTER 3 Probability Extending Concepts According to Bayes’ Theorem, the probability of event A, given that event B has occurred, is P1A B2 = P1A2 # P1B A2 P1A2 # P1B A2 + P1A′2 # P1B A′2 . In Exercises 33–38, use Bayes’ Theorem to find P1A B2. 33. P1A2 = 2 3, P1A′2 = 1 3, P1B A2 = 1 5, and P1B A′2 = 1 2 34. P1A2 = 3 8, P1A′2 = 5 8, P1B A2 = 2 3, and P1B A′2 = 3 5 35. P1A2 = 0.25, P1A′2 = 0.75, P1B A2 = 0.3, and P1B A′2 = 0.5 36. P1A2 = 0.62, P1A′2 = 0.38, P1B A2 = 0.41, and P1B A′2 = 0.17 37. P1A2 = 73%, P1A′2 = 17%, P1B A2 = 46%, and P1B A′2 = 52% 38. P1A2 = 12%, P1A′2 = 88%, P1B A2 = 66%, and P1B A′2 = 19% 39. Reliability of Testing A virus infects one in every 200 people. A test used to detect the virus in a person is positive 80% of the time when the person has the virus and 5% of the time when the person does not have the virus. (This 5% result is called a false positive.) Let A be the event “the person is infected” and B be the event “the person tests positive.” (a) Using Bayes’ Theorem, when a person tests positive, determine the probability that the person is infected. (b) Using Bayes’ Theorem, when a person tests negative, determine the probability that the person is not infected. 40. Birthday Problem You are in a class that has 24 students. You want to find the probability that at least two of the students have the same birthday. (a) Find the probability that each student has a different birthday. (b) Use the result of part (a) to find the probability that at least two students have the same birthday. (c) Use technology to simulate the “Birthday Problem” by generating 24 random numbers from 1 to 365. Repeat the simulation 10 times. How many times did you get at least two people with the same birthday? The Multiplication Rule and Conditional Probability By rewriting the formula for the Multiplication Rule, you can write a formula for finding conditional probabilities. The conditional probability of event B occurring, given that event A has occurred, is P1B A2 = P1A and B2 P1A2 . In Exercises 41 and 42, use the information below. • The probability that an airplane flight departs on time is 0.89. • The probability that a flight arrives on time is 0.87. • The probability that a flight departs and arrives on time is 0.83. 41. Find the probability that a flight departed on time given that it arrives on time. 42. Find the probability that a flight arrives on time given that it departed on time.
RkJQdWJsaXNoZXIy NjM5ODQ=