Question

Question 1: Calculation (10 marks)

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 (5 marks)

A chair is randomly selected.

(b) What is the probability that the chair that robot Q made is defective (1 marks)

(c) What is the probability of findings a broken chair (2 marks)

(d) Given that a chair is defective, what is the probability that it was not made by robot R (2 marks)

Answer 1

(a)

(b)

$$P(D|Q)=0.02$$

(c)

By the Law of Total Probability

$$P(D)=P(Q)P(D|Q)+P(R)P(D|R)+P(S)P(D|S)$$

$$=0.25(0.02)+0.45(0.03)+0.30(0.05)=0.0335$$

(d)

By Bayes' Theorem

$$P(R'|D)=\dfrac{P(Q)P(D|Q)+P(S)P(D|S)}{P(Q)P(D|Q)+P(R)P(D|R)+P(S)P(D|S)}$$

$$=\dfrac{0.25(0.02)+0.30(0.05)}{0.25(0.02)+0.45(0.03)+0.30(0.05)}$$

$$=\dfrac{0.02}{0.0335}=\dfrac{40}{67}\approx0.597015$$