I. The probability that your call to a service line is answered in less than 30
ID: 3127448 • Letter: I
Question
I. The probability that your call to a service line is answered in less than 30 seconds is 0.65. Assume that your calls are independent.
(a) If you call 13 times, what is the probability that exactly 9 of your calls are answered within 30 seconds? Round your answer to four decimal places
(b) If you call 20 times, what is the probability that at least 16 calls are answered in less than 30 seconds? Round your answer to four decimal places
(c) If you call 20 times, what is the mean number of calls that are answered in less than 30 seconds? Round your answer to the nearest integer.
II. Astronomers treat the number of stars in a given volume of space as a Poisson random variable. The density in the Milky Way Galaxy in the vicinity of our solar system is 1 star per 16 cubic light years.
(a) What is the probability of 3 or more stars in 16 cubic light years?
(b) How many cubic light years of space must be studied so that the probability of 1 or more stars exceeds 0.96?
Explanation / Answer
I.
a)
Note that the probability of x successes out of n trials is
P(n, x) = nCx p^x (1 - p)^(n - x)
where
n = number of trials = 13
p = the probability of a success = 0.65
x = the number of successes = 9
Thus, the probability is
P ( 9 ) = 0.222227822 [ANSWER]
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b)
Note that P(at least x) = 1 - P(at most x - 1).
Using a cumulative binomial distribution table or technology, matching
n = number of trials = 20
p = the probability of a success = 0.65
x = our critical value of successes = 16
Then the cumulative probability of P(at most x - 1) from a table/technology is
P(at most 15 ) = 0.881803441
Thus, the probability of at least 16 successes is
P(at least 16 ) = 0.118196559 [ANSWER]
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c)
Mean = n p = 20*0.65 = 13 [ANSWER]
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