Write clearly, please. Answer all parts. Show all work and steps [1] A manufactu

Question

Write clearly, please. Answer all parts. Show all work and steps

[1] A manufacturing company manufactures 100 products every day. There is a 5% chance that any of the manufactured products is defective and the products are independent. Consider the 100 products manufactured on a single day. Find the probability that (a) only the first and the last products are defective and the rest are good (b) no more than 2 products are defective (c) the third defective product happens to be the 60h product (d) there are two defective products within the first 50 products and two more defective products within the second set of 50 products.

Explanation / Answer

Solution

Given

Number of products, n = 100

P(defective) = 0.05 => P(good) = 0.95

Products are independent with respect to defect. => probability

=> P(k products are defective) = 0.05k and

P(k1 products are defective and k2 products are good) = 0.05k1 x 0.95k2

Part (a)

Required probability

= P(first product is defective) x P(next 98 products are good) x P(last product is defective)

= 0.05 x 0.9598 x 0.05

= 1.64004E-05 ANSWER

Part (b)

Probability no more than two products are defective

= Probability a maximum of two products are defective

= P(0 defective) + P(1 defective) + P(2 defectives)

= P(X = 0) + P(X = 1) + P(X = 2), where X = number of defectives out of 100 products.

We know that X ~ B(100, 0.05) and hence the above probability can be found using Excel Function on Binomial Distribution as follows:

= 0.005921 + 0.031161 + 0.081182

= 0.118263 ANSWER

Part (c)

Third defective product is the 60th product

=> there are only 2 defectives among the first 59 products and the 60th product is defective.

Hence, the required probability

= P(There are only 2 defectives among the first 59 products) x P(60th product is defective)

= P(Y = 2) x 0.05, where Y = number of defectives among the first 59 products ~ B(59, 0.05)

= 0.229845 x 0.05 [using Excel Function on Binomial Distribution]

= 0.011492 ANSWER

Part (d)

Required probability

= P(Z = 2) x P(W = 2), where Z = number of defectives among the first 50 products and W = number of defectives among the next 50 products, both of which have B(50, 0.05)

= 0.261101 x 0.261101

= 0.068174 ANSWER

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