Cloth is sold by the bolt where a bolt is 15 yards of material. If the expected number of blemishes per yard is 0.1 find the probability tha

Question

Cloth is sold by the bolt where a bolt is 15 yards of material. If the expected number of blemishes per yard is 0.1 find the probability that:

(a) A yard of cloth contains 1 or more blemishes.
(b) A bolt contains at most 2 blemishes.
(c) A lot of 30 bolts contains at least 50 blemishes.

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Delilah 1 month 2021-11-08T11:21:01+00:00 1 Answer 0 views 0

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    2021-11-08T11:22:01+00:00

    Answer:

    (a) The probability that a yard of cloth contains 1 or more blemishes is 0.0952.

    (b) The probability that a yard of cloth contains at most 1 blemishes is 0.9953.

    (c) The probability that in a lot of 30 bolts contains at least 50 blemishes is 0.2468.

    Step-by-step explanation:

    Let X = number of blemishes per yard of material.

    The expected value of X is:

    E (X) = λ = 0.1

    The random variable  X follows a Poisson distribution with parameter λ = 0.10

    The probability function of a Poisson distribution is:

    P(X=x)=\frac{e^{-\lambda}\lambda^{x}}{x!} ;\ x=0, 1, 2, ...

    (a)

    Compute the probability that a yard of cloth contains 1 or more blemishes as follows:

    P (X ≥ 1) = 1 – P (X < 1)

                 = 1 – P (X = 0)

                 =1-\frac{e^{-0.10}(0.10)^{0}}{0!}\\=1-0.90484\\=0.09516\\\approx0.0952

    Thus, the probability that a yard of cloth contains 1 or more blemishes is 0.0952.

    (b)

    Compute the probability that a yard of cloth contains at most 1 blemishes as follows:

    P (X ≤ 1) = P (X = 0) + P (X = 1)

                 =\frac{e^{-0.10}(0.10)^{0}}{0!}+\frac{e^{-0.10}(0.10)^{1}}{1!}\\=0.90484+0.090484\\=0.995324\\\approx0.9953

    Thus, the probability that a yard of cloth contains at most 1 blemishes is 0.9953.

    (c)

    In a lot of 30 bolts the expected number of yards of material is:

    \lambda_{1}=0.10\times30\times15=45

    Compute the probability that in a lot of 30 bolts contains at least 50 blemishes as follows:

    P (X ≥ 50) = 1 – P (X < 50)

                    =1-0.7532\\=0.2468

    **Use an online calculator or Excel to compute the probability.

    Thus, the probability that in a lot of 30 bolts contains at least 50 blemishes is 0.2468.

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