Question 3: You are hired by a Company that manufactures microprocessors as a qu

Question

Question 3: You are hired by a Company that manufactures microprocessors as a quality assurance engineer. Your duty is to monitor 4 manufacturing machines. The following table describes the manufacturing processor. Machine Number Manufactured microprocessors Percentage of being within / Hour 50KHz of processing speed quality objectives 80% 85% M1 M2 M3 M4 4000 4500 1500 2000 90% Every hour all microprocessors from the four machines are mixed randomly and packed in one box and shipped overseas. Based on the above table, answer each of the following questions

Explanation / Answer

Let M1 shows the event that microprocessor manufactured by machine M1, M2 shows the event that microprocessro manufcatured by machine 2, M3 shows the event that microprocessor manufactured by machine 3 and M4 shows the event that microprocessor manufactured by machine 4. Let S shows the event that microprocessor is under specification.

Total number of micorprocessor manufcatured every hour is : 4000 +4500 +1500 +2000 = 12000

So,

P(M1) = 4000 /12000 = 0.3333

P(M2) = 4500 /12000 = 0.3750

P(M3) = 1500 /12000 = 0.1250

P(M4) = 2000 /12000 = 0.1667

From table we have

P(S|M1) = 0.80, P(S|M2) = 0.85, P(S|M3) = 0.70, P(S|M4) = 0.90

a)

By the law of total probability, the probability that microprocessor is meeting quality objectives is

P(S) = P(S|M1)P(M1)+ P(S|M2)P(M2)+ P(S|M3)P(M3)+ P(S|M4)P(M4) = 0.80 * 0.3333 + 0.85 * 0.375 + 0.70 * 0.125 + 0.90 * 0.1667 = 0.82292

The required probability is 0.8229.

(b)

The required probability is

P(M3|S) =[ P(S|M3)P(M3)] / P(S) = 0.0875 / 0.8229 = 0.1063

(c)

Since company M3 has lowest chance to produce the microprocessor to meet the objectives so except M3 manager can increase the production fro rest three machines. Following table shows a possible solution with corresponding probabilities:

In the above scenario, the required probability is 85.1% which meets the requirement.

Mi P(Mi) P(S|Mi) P(S|Mi)P(Mi) 5000 0.208333 0.8 0.166666667 7500 0.3125 0.85 0.265625 1500 0.0625 0.7 0.04375 10000 0.416667 0.9 0.375 Total 24000 0.851041667

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