Traffic engineers install 10 street lights with new bulbs. The probability that
ID: 3130209 • Letter: T
Question
Traffic engineers install 10 street lights with new bulbs. The probability that a bulb will fail within 50020 hours of operation is 0.27. Assume that each of the bulbs fails independently.
(a) What is the probability that fewer than two of the original bulbs will fail within 50020 hours of operation?
(b) What is the probability that no bulbs will have to be replaced within 50020 hours of operation?
(c) What is the probability that more than four of the original bulbs will need replacing within 50020 hours?
Explanation / Answer
a)
Note that P(fewer than x) = P(at most x - 1).
Using a cumulative binomial distribution table or technology, matching
n = number of trials = 10
p = the probability of a success = 0.27
x = our critical value of successes = 2
Then the cumulative probability of P(at most x - 1) from a table/technology is
P(at most 1 ) = 0.201929542
Which is also
P(fewer than 2 ) = 0.201929542 [ANSWER]
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b)
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 = 10
p = the probability of a success = 0.27
x = the number of successes = 0
Thus, the probability is
P ( 0 ) = 0.042976258 [ANSWER]
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c)
Note that P(more than x) = 1 - P(at most x).
Using a cumulative binomial distribution table or technology, matching
n = number of trials = 10
p = the probability of a success = 0.27
x = our critical value of successes = 4
Then the cumulative probability of P(at most x) from a table/technology is
P(at most 4 ) = 0.896316856
Thus, the probability of at least 5 successes is
P(more than 4 ) = 0.103683144 [ANSWER]
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