Question

A manufacturer obtains​ clock-radios from three different​ subcontractors: 30​% from​ A, 30​% from​ B, and 40​% from C. The defective rates for these subcontractors are 1​%, 5%​, and1​% respectively. If a defective​ clock-radio is returned by a​ customer, what is the probability that it came from subcontractor​ A? From​ B? From​ C?

Answer 1

To find the probability that a defective clock-radio came from a particular subcontractor, we can use Bayes' theorem. The probability that a clock-radio came from subcontractor A given that it is defective can be calculated as 9.375%. The probabilities for subcontractors B and C can also be calculated using Bayes' theorem.

Step-by-step explanation:

To find the probability that a returned defective clock-radio came from subcontractor A, we need to use Bayes' theorem. The probability of an event A given an event B is equal to the probability of event B given event A multiplied by the probability of event A, divided by the probability of event B.

Let's denote the events as follows:

A: Defective clock-radio came from subcontractor A

B: Defective clock-radio is returned

P(A) = 0.30 (30% of clock-radios are from subcontractor A)

P(B|A) = 0.01 (defective rate of subcontractor A)

P(B) = P(B|A) * P(A) + P(B|B) * P(B) + P(B|C) * P(C)

P(B) = 0.01 * 0.30 + 0.05 * 0.30 + 0.01 * 0.40 = 0.032

To find the probability that a defective clock-radio came from subcontractor A, we use Bayes' theorem:

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

P(A|B) = (0.01 * 0.30) / 0.032 = 0.09375 (or 9.375%)

Similarly, we can calculate the probabilities for subcontractor B and C:

P(B|B) = (0.05 * 0.30) / 0.032 = 0.46875 (or 46.875%)

P(C|B) = (0.01 * 0.40) / 0.032 = 0.125 (or 12.5%)

Answer 2

Using Bayes' theorem, the probability that a defective clock-radio came from subcontractor A is approximately 13.64%, from subcontractor B is approximately 68.18%, and from subcontractor C is approximately 18.18%.

Bayes' theorem describes the conditional probability of an event occurring given another condition.

Let event A = clock-radio came from subcontractor A

Let event B = clock-radio came from subcontractor B

Let event C = clock-radio came from subcontractor C

Let event D = clock-radio is defective

Given:

P(A) = 30% = 0.30

P(B) = 30% = 0.30

P(C) = 40% = 0.40

P(D|A) = 1% = 0.01

P(D|B) = 5% = 0.05

P(D|C) = 1% = 0.01

P(A|D) = the probability that a defective clock-radio came from subcontractor A

P(B|D) = probability that a defective clock-radio came from subcontractor B

P(C|D) = probability that a defective clock-radio came from subcontractor C

Using Bayes' theorem:

P(A|D) = (P(D|A) * P(A)) / (P(D|A) * P(A) + P(D|B) * P(B) + P(D|C) * P(C))

= (0.01 * 0.30) / (0.01 * 0.30 + 0.05 * 0.30 + 0.01 * 0.40)

= 0.003 / (0.003 + 0.015 + 0.004)

= 0.003 / 0.022

≈ 0.1364

P(B|D) = (P(D|B) * P(B)) / (P(D|A) * P(A) + P(D|B) * P(B) + P(D|C) * P(C))

= (0.05 * 0.30) / (0.01 * 0.30 + 0.05 * 0.30 + 0.01 * 0.40)

= 0.015 / (0.003 + 0.015 + 0.004)

= 0.015 / 0.022

≈ 0.6818

P(C|D) = (P(D|C) * P(C)) / (P(D|A) * P(A) + P(D|B) * P(B) + P(D|C) * P(C))

= (0.01 * 0.40) / (0.01 * 0.30 + 0.05 * 0.30 + 0.01 * 0.40)

= 0.004 / (0.003 + 0.015 + 0.004)

= 0.004 / 0.022

≈ 0.1818

Thus, the probability that a defective clock-radio came from subcontractor A is approximately 13.64%, from subcontractor B is approximately 68.18%, and from subcontractor C is approximately 18.18%.