Answer to Question #245578 in Statistics and Probability for Kamal

Question

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

 

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

Answer 1

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) $=0.005$

Answer = 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) = (0.25x0.02)+(0.45x0.03)+(0.30x0.05)

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