Question

The foreman in an iron foundry knows that the sand used for molding iron castings is too dry 4%4% of the time and too wet 4%4% of the time. He also knows that defective castings occur 0.13%0.13% of the time when the sand has the correct amount of moisture; 4.7%4.7% of the time when the sand is too dry; and 17.5%17.5% of the time when the sand is too wet. Suppose that a casting is selected at random from the batch just produced. Please provide a detailed solution with four decimals.

(a) What is the probability that the molding is good?

(b) If the molding is good, what is the probability that the sand is too wet?

(c) If the molding is good, what is the probability that the moisture content of the sand is correct?

Answer 1

Given data

P(too dry) = 0.04 ,P(too wet) = 0.04

P(correct) = 1 - 0.04 -0.04 = 0.920

Let D shows the event of defective casting. So

P(defective casting| too dry) = 0.047

P(defective casting| too wet) = 0.175

P(defective casting| correct) = 0.0013

By the complement rule we have

P(good casting| too dry)=1- P(defective casting| too dry) =1- 0.047 = 0.953

P(good casting| too wet)=1- P(defective casting| too wet) =1- 0.175 = 0.825

P(good casting| correct)=1-P(defective casting| correct) = 1-0.0013 = 0.9987

(a)

By the law of total probability we have

P(good casting) =

P(good casting| too dry)P(too dry) + P(good casting| too wet)P(too wet) + P(good casting| correct) P(correct)

= 0.953*0.04 + 0.825*0.04 + 0.9987*0.920 = 0.98992

So required probability is 0.9899

(b)

The required probability is

P(too wet| good casting) = [P(good casting| too wet)P(too wet)] / P(good casting)

= [0.825*0.04] / 0.9899 = 0.0333

(c)

The required probability is

P(correct| good casting) = [P(good casting| correct)P(correct)] / P(good casting)

= [0.9987*0.920] / 0.9899 = 0.9282