Question

Suppose a batch of widgets has a 50% prior probability of coming from supplier 1; otherwise it comes from supplier 2. Suppose 3% of supplier 1’s widgets are defective and 2% of supplier 2’s widgets are defective (independently). Ten widgets from the batch are tested.

a. What is the probability the first widget is defective?

b. What is the probability the second widget is defective?

c. What is the probability the second widget is defective given the first is defective?

d. What is the probability that the first widget is defective given the 10th widget is defective?

e. What is the probability the first widget is defective if we know it came from supplier 1?

f. What is the probability the second widget is defective if the first is defective and we know the batch came from supplier 1?

g. What are the posterior odds the batch came from supplier 1 given two widgets are tested and found defective?

h. What is the posterior probability (write the expression) that the batch is from supplier 1 if all ten widgets are defective?

Answer 1

Hello Sir/ Mam

We are most delighted to answer, but I would request you to please ask atmost 1 question or 4 subparts per post. Your cooperation is highly appreciated. Thanks!

I'll be answering Parts (a) to (d) here. Please consider creating a fresh post for the remaining parts specifically asking to solve the remaining parts. We will be happy to serve you again. Thanks!

Let S1 be the event that the widget came from supplier 1.

Let S2 be the event that the widget came from supplier 2.

Let D_i be the event that the i^th widget is defective.

Hence,

(a)

(b)

(c)

(BECAUSE D₂ and D₁ are independent)

(d)

(BECAUSE D₁ and D₁₀ are independent)

I hope this solves your doubt.

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