Question

A) Machine A produces an average of 10% of defective parts. Machine B produces on average twice as many defective parts as machine A. Machine C produces on average three times more defective parts than machine B. Here are 8 bags of parts from A, 2 bags of parts from B and 1 bag of coins from C. We randomly take a coin from one of the bags itself also taken at random. She is defective.

1. What is the probability that this piece comes from B?
2. What is the probability that this piece does not come from A?

B) A STOP sign has been placed at an extremely dangerous crossroads where a small road leads to a very busy highway. Statistical data indicates that 20% of motorists do not comply not this STOP sign and that 30% of motorists who do not respect the STOP sign have an accident crossroads. The same data also indicate that 95% of motorists complying with this STOP sign have no accident at this crossroads. We consider a random motorist arriving at this crossroads, to determine the probability of the following events:

1. the motorist has an accident at this crossroads,
2. knowing that the motorist has respected the STOP sign, what is the probability that he has an accident crossroads ?

Answer 1

Let A, B and C denote the event of selecting parts from Machine A, Machine B and Machine C respectively

Let D denote the event of producing defective parts by a machine

P(A) = 8/(8 + 2 + 1) = 8/11

P(B) = 2/(8 + 2 + 1) = 2/11

P(C) = 1/(8 + 2 + 1) = 1/11

P(D | A) = 0.10

P(D | B) = 2P(D | A) = 0.20

P(D | C) = 3P(D | B) = 0.60

Thus, P(D) = 0.10*8/11 + 0.20*2/11 + 0.60*1/11 = 9/55

(1) The required probability = P(B | D)

= P(D | B)*P(B)/P(D)

= 2/9

(2) The required probability = P(A' | D)

= 1 - P(A | D)

= 1 - 0.10*8/(9/55)

= 5/9

B)

Let M and A denote the events that the motorists does not comply with the STOP sign and have an accident crossroads respectively

Thus, P(M) = 0.20, P(M') = 0.80

P(A | M) = 0.30

P(A' | M') = 0.95 -> P(A | M') = 0.05

1) Probability that a randomly selected motorist has an accident at this crossroads = P(A)

= P(A | M)*P(M) + P(A | M')*P(M')

= 0.30*0.20 + 0.05*0.80

= 0.10

2) The required probability = P(A | M') = 0.05