# Answer in Statistics and Probability for Kamal #245578

In a workshop, three robots, Q, R and S, are employed to make chairs

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Robot Q makes 25% of the chairs

Robot R makes 45% of the chairs

The remaining chairs are made by Robot S

Evidence has shown that 2 percent of the chairs made by robot Q are defective, 3 percent of the chairs made by robot R, and 5 percent of the chairs made by robot S are defective

(a) Construct a tree diagram that illustrates all possible outcomes and probabilities

A chair is randomly selected.

(b) What is the probability that the chair that robot Q made is defective

(c) What is the probability of findings a broken chair

(d) Given that a chair is defective, what is the probability that it was not made by robot R

SOLUTION

(a) Construct a tree diagram that illustrates all possible outcomes and probabilities

(b) What is the probability that the chair that robot Q made is defective

Solution:

By visual inspection using the Decision Tree

P(Defective “\bigcap” Robot Q) = P(Robot Q) x P(Defective | Robot Q)

P(Defective “\bigcap” Robot Q) “=0.25*0.02”

P(Defective “\bigcap” Robot Q) “=0.005”

(c) What is the probability of findings a broken chair

By visual inspection using the Decision Tree

P(Defective) = P(Defective Robot Q) + P(Defective Robot R) + P(Defective Robot S)

P(Defective) = (0.25×0.02)+(0.45×0.03)+(0.30×0.05)

P(defective) = (0.005)+(0.0135)+(0.015)

P(defective) = 0.0335

(d) Given that a chair is defective, what is the probability that it was not made by robot R

From (c) we know that P(Defective) = 0.0335, then by definition:

The probability that the defective chair was made by robot R is:

P(Robot R | Defective) = [P(Robot R) x P(Defective | Robot R] / P(Defective)

P(Robot R | Defective) =[(.45)x(0.03)] / (0.0335)

P(Robot R | Defective) =0.0135 / 0.0335

P(Robot R | Defective) =0.4030

However, we are looking for the probability that a random defective chair is not made by Robot R. Then we have:

P(not Robot R | Defective)= 1 – P(Robot R | Defective)

P(not Robot R | Defective)= 0.5970

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