答案和详解如下： 6、Given the following table about employees of a company based on whether they are smokers or nonsmokers and whether or not they suffer from any allergies, what is the probability of both suffering from allergies and not suffering from allergies?
| Suffer from Allergies | Don't Suffer from Allergies | Total | Smoker | 35 | 25 | 60 | Nonsmoker | 55 | 185 | 240 | Total | 90 | 210 | 300 |
A) 0.50. B) 0.00. C) 1.00. D) 0.24. The correct answer was B) These are mutually exclusive, so the joint probability is zero. 7、In a given portfolio, half of the stocks have a beta greater than one. Of those with a beta greater than one, a third are in a computer-related business. What is the probability of a randomly drawn stock from the portfolio having both a beta greater than one and being in a computer-related business? A) 0.333. B) 0.167. C) 0.500. D) 0.667. The correct answer was B) This is a joint probability. From the information: P(beta > 1) = 0.500 and P(comp.stock|beta > 1) = 0.333. Thus, the joint probability is the product of these two probabilities: (0.500)*(0.333) = 0.167. 8、A parking lot has 100 red and blue cars in it. §
40% of the cars are red. §
70% of the red cars have radios. §
80% of the blue cars have radios. What is the probability of selecting a car at random and having it be red and have a radio? A) 25%. B) 28%. C) 40%. D) 48%. The correct answer was B) Joint probability is the probability that both events, in this case a car being red and having a radio, happen at the same time. Joint probability is computed by multiplying the individual event probabilities together: P(red and radio) = (P(red)) * (P(radio)) = (0.4) * (0.7) = 0.28 or 28 percent.
| Radio | No Radio |
| Red | 28 | 12 | 40 | Blue | 48 | 12 | 60 |
| 76 | 24 | 100 |
9、What is the probability of selecting a car at random that is either red or has a radio? A) 88%. B) 28%. C) 76%. D) 116%. The correct answer was A) The addition rule for probabilities is used to determine the probability of at least one event among two or more events occurring, in this case a car being red or having a radio. To use the addition rule, the probabilities of each individual event are added together, and, if the events are not mutually exclusive, the joint probability of both events occurring at the same time is subtracted out: P(red or radio) = P(red) + P(radio) – P(red and radio) = 0.40 + 0.76 – 0.28 = 0.88 or 88 percent. 10、What is the probability that the car is red given that you already know that it has a radio? A) 28%. B) 47%. C) 37%. D) 88%. The correct answer was C) Given a set of prior probabilities for an event of interest, Bayes’ formula is used to update the probability of the event, in this case that the car we already know has a radio is red. Bayes’ formula says to divide the Probability of New Information given Event by the Unconditional Probability of New Information and multiply that result by the Prior Probability of the Event. In this case, P(red car has a radio) = 0.70 is divided by 0.76 (which is the Unconditional Probability of a car having a radio (40 percent are red of which 70 percent have radios) plus (60 percent are blue of which 80 percent have radios) or ((0.40) * (0.70)) + ((0.60) * (0.80)) = 0.76.) This result is then multiplied by the Prior Probability of a car being red, 0.40. The result is (0.70/0.76) * (0.40) = 0.37 or 37 percent. | 
发表于 2008-4-9 15:11
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