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Chapter 3: Q82E (page 200)
Suppose the events are mutually exclusive and complementary events, such that, and . Consider another event A such thatandUse Baye鈥檚 Rule to find
a.
b.
c.role="math" localid="1658214716845"
Therefore, the values of all parts are:
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Ambulance response time.Geographical Analysis(Jan. 2010) presented a study of emergency medical service (EMS) ability to meet the demand for an ambulance. In one example, the researchers presented the following scenario. An ambulance station has one vehicle and two demand locations, A and B. The probability that the ambulance can travel to a location in under 8 minutes is .58 for location A and .42 for location B. The probability that the ambulance is busy at any point in time is .3.
a.Find the probability that EMS can meet the demand for an ambulance at location A.
b.Find the probability that EMS can meet the demand for an ambulance at location B.
Reliability of gas station air gauges. Tire and automobile manufacturers and consumer safety experts all recommend that drivers maintain proper tire pressure in their cars. Consequently, many gas stations now provide air pumps and air gauges for their customers. In a Research Note(Nov. 2001), the National Highway Traffic Safety Administration studied the reliability of gas station air gauges. The next table gives the percentage of gas stations that provide air gauges that over-report the pressure level in the tire.
a. If the gas station air pressure gauge reads 35 psi, what is the probability that the pressure is over-reported by 6 psi or more?
b. If the gas station air pressure gauge reads 55 psi, what is the probability that the pressure is over-reported by 8 psi or more?
c. If the gas station air pressure gauge reads 25 psi, what is the probability that the pressure is not over-reported by 4 psi or more?
d. Are the events A= {over report by 4 psi or more} and B= {over report by 6 psi or more} mutually exclusive?
e.Based on your answer to part d, why do the probabilities in the table not sum to 1?
Question: Refer to Exercise 3.35. Use the same event definitions to do the following exercises.
a. Write the event that the outcome is "On" and "High" as an intersection of two events.
b. Write the event that the outcome is "Low" or "Medium" as the complement of an event.
Which events are independent?Use your intuitive understanding of independence to form an opinion about whether each of the following scenarios represents independent events.
a.The results of consecutive tosses of a coin.
b.The opinions of randomly selected individuals in a pre-election poll.
c.A Major League Baseball player's results in two consecutive at-bats.
d.The amount of gain or loss associated with investments in different stocks if these stocks are bought on the same day and sold on the same day 1 month later.
e.The amount of gain or loss associated with investments in different stocks bought and sold in different time periods, 5 years apart.
f.The prices bid by two different development firms in response to a building construction proposal.
The three-dice gambling problem. According toSignificance(December 2015), the 16th-century mathematician Jerome Cardan was addicted to a gambling game involving tossing three fair dice. One outcome of interest鈥� which Cardan called a 鈥淔ratilli鈥濃�攊s when any subset of the three dice sums to 3. For example, the outcome {1, 1, 1} results in 3 when you sum all three dice. Another possible outcome that results in a 鈥淔ratilli鈥� is {1, 2, 5}, since the first two dice sum to 3. Likewise, {2, 3, 6} is a 鈥淔ratilli,鈥� since the second die is a 3. Cardan was an excellent mathematician but calculated the probability of a 鈥淔ratilli鈥� incorrectly as 115/216 = .532.
a. Show that the denominator of Cardan鈥檚 calculation, 216, is correct. [Hint: Knowing that there are 6 possible outcomes for each die, show that the total number of possible outcomes from tossing three fair dice is 216.]
b. One way to obtain a 鈥淔ratilli鈥� is with the outcome {1,1, 1}. How many possible ways can this outcome be obtained?
c. Another way to obtain a 鈥淔ratilli鈥� is with an outcome that includes at least one die with a 3. First, find the number of outcomes that do not result in a 3 on any of the dice. [Hint: If none of the dice can result in a 3, then there are only 5 possible outcomes for each die.] Now subtract this result from 216 to find the number of outcomes that include at least one 3.
d. A third way to obtain a 鈥淔ratilli鈥� is with the outcome {1, 2, 1}, where the order of the individual die outcomes does not matter. How many possible ways can this outcome be obtained?
e. A fourth way to obtain a 鈥淔ratilli鈥� is with the outcome {1, 2, 2}, where the order of the individual die outcomes does not matter. How many possible ways can this outcome be obtained?
f. A fifth way to obtain a 鈥淔ratilli鈥� is with the outcome {1, 2, 4}, where the order of the individual die outcomes does not matter. How many possible ways can this outcome be obtained? [Hint:There are 3 choices for the first die, 2 for the second, and only 1 for the third.]
g. A sixth way to obtain a 鈥淔ratilli鈥� is with the outcome {1, 2, 5}, where the order of the individual die outcomes does not matter. How many possible ways can this outcome be obtained? [See Hintfor part f.]
h. A final way to obtain a 鈥淔ratilli鈥� is with the outcome {1, 2, 6}, where the order of the individual die outcomes does not matter. How many possible ways can this outcome be obtained? [See Hintfor part f.]
i. Sum the results for parts b鈥揾 to obtain the total number of possible 鈥淔ratilli鈥� outcomes.
j. Compute the probability of obtaining a 鈥淔ratilli鈥� outcome. Compare your answer with Cardan鈥檚.
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