George, Richard, Paul, and John play the following game. Each man takes one of four balls numbered 1 through 4 from an urn. The man who draws ball 4 loses. The other three return their balls to the urn and draw again. Now the one who draws ball 3 loses. The other two return their balls to the urn and draw ball 3 loses. The man who draws ball 1 wins the game.
- What is the probability that John does not lose in the first two draws?
- What is the probability that Paul wins the game?
The health department routinely conducts independent inspections of each restaurant, with the restaurant passing only if both inspectors pass it. Inspector A is very experienced, and hence, passes only 2 prevent of restaurants that actually do have code violations. Inspector B is less experienced and passes 7 percent of restaurants with violations. What is the probability that
- Inspector A passes a restaurant, given that inspector B has found a violation?
- Inspector B passes a restaurant with a violation, given that inspector A passes it?
- A restaurant with a violation is passed by the health department?
The four floodgates of a small hydroelectric dam fail and are repaired independently of each other. From experience, it's known that each floodgate is out of order 4 percent of the time.
- If floodgate 1 is out of order, what is the probability that floodgates 2 and 3 are out of order?
- During a tour of the dam, you are told that the chances of all four floodgates being out of order are less than 1 in 5,000,000. Is this statement true?
Rob Rales is preparing a report that his employer, the titre Corporation, will eventually deliver to the Federal Aviation Administration. First, the report must be approved by Rob's group leader, department head, and division chief. Rob knows from experience that the three managers act independently. Further, he knows that his groups leader approves 85 percent of his reports, his department head approves 80 percent of the reports written by Rob that each him, and his division chief approves 82 percent of Rob"s work.
- What is the probability thaty he first version of Rob's report is submitted to the FAA?
- What is the probability that the first version of Rob's report is approved by this group leader and department head, but is not approved by his division chief?
A grocery store is reviewing its restocking policies and has analyzed the number of half-gallon containers of orange juice sold each day for the past month. The data are given below.
- What is the probability that on a randomly selected day the number of cartons of orange juice sold in the evening is between 80 and 99?
- What is the probability that 39 or fewer cartons were sold during a randomly selected afternoon?
- What is the probability that either 0-19 or 100 or more cartons were sold in a randomly selected morning?
Bill Borde, top advertising executive for Grapevine Concepts, has just launched a publicity campaign for a new restaurant in town, The Black Angus. Bill has just installed four billboards on a highway outside of town, and he knows from experience the probabilities that each will be noticed by a randomly chosen motorist. The probability of the first billboard's being noticed by a motorist is 0.75. The probability of the second being noticed is 0.82, the third has a probability of 0.87 of being noticed, and the probability of the fourth sign's being noticed is 0.9. Assuming that the event that a motorist noticed any particular billboard is independent of whether or not he notices the others, what is the probability that.
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According to a survey, the probability that a family owns two cars if its annual income is greater then $35000 is 0.75. Of the households surveyed, 60 percent had incomes over $35000 and 52 percent had two cars. What is the probability that a family has two cars and an income over $35,000 a year?
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Friendly's Department Store has been the target of many shoplifters during the past month, but owing to increased security precautions, 250 shoplifters have been caught. Each shoplifter's sex is noted; also noted is whether the perpetrator was a first time or repeat offender. The data are summarized in the table.
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Two events, A and B, are statistically dependent. If P(A)=0.39, P(B)=0.21, and P(A or B)=0.47, find the probability that
- Neither A nor B will occur.
- Both A and B will occur.
- B will occur, given that A has occurred.
- A will occur, given that B has occurred
Given that P(A) = 3/14, P(B) = 1/3, P(AC) = 1/7 , and P(B|C) = 5/21, find the following probabilities: P(A|C), P(C|A), P(BC), P(C|B).
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