The number of surface flaws in a plastic roll used in the interior of automobile
ID: 3128146 • Letter: T
Question
The number of surface flaws in a plastic roll used in the interior of automobiles has a Poisson distribution with a mean of 0.05 flaw per square foot of plastic roll. Assume an automobile interior contains 8 square feet of plastic roll. Round your answers to four decimal places (e.g. 98.7654).
(a) What is the probability that there are no surface flaws in an auto’s interior?
(b) If 16 cars are sold to a rental company, what is the probability that none of the 16 cars has any surface flaws?
(c) If 16 cars are sold to a rental company, what is the probability that at most 4 cars has any surface flaws?
Explanation / Answer
a)
Note that the probability of x successes out of n trials is
P(x) = u^x e^(-u) / x!
where
u = the mean number of successes = 0.05*8 = 0.4
x = the number of successes = 0
Thus, the probability is
P ( 0 ) = 0.670320046 [ANSWER]
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b)
That is, all 16 have no flaw.
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 = 16
p = the probability of a success = 0.670320046
x = the number of successes = 16
Thus, the probability is
P ( 16 ) = 0.001661557 [ANSWER]
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c)
That is, at least 12 have no flaws.
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 = 16
p = the probability of a success = 0.670320046
x = our critical value of successes = 12
Then the cumulative probability of P(at most x - 1) from a table/technology is
P(at most 11 ) = 0.649395891
Thus, the probability of at least 12 successes is
P(at least 12 ) = 0.350604109 [ANSWER]