Problem 2 The vaccine can be purified by one of three different machines:Machine

Question

Problem 2

The vaccine can be purified by one of three different machines:Machine A, Machine B or Machine C.

Of 1200 products sampled, 420 were processed on Machine A, 510 wereprocessed on Machine B and the remainder on Machine C.

The quality control department would like to know the overallprobability of producing a defective product. Use a table to findthe probability of a defect at this step.

Write all intermediate steps and formulas used. Show all work forfull credit.
[Hint: First put the above information into the propernotation. Next, set up a probability tree or table, fill in thegiven information and find the missing information.]

1. Set up a table (or probabilitytree) to answer the questions below.
2. What is the probability ofproducing a defect at the purification step?
3. Given that there is a defect,what is the probability that it was processed on:
   • Machine A?  
     
   • MachineB?   
     
   • Machine C?

Explanation / Answer

P(defect)=(1/3)*0.005+(1/3)*0.01+(1/3)*0.05 = 0.0217 (1) Machine A: P(Machine A | defect)=P(Machine A and defect)/P(defect)                                =(1/3)*0.005/0.0217                                =0.0768 (2) Machine B: P(Machine B | defect)=P(Machine B and defect)/P(defect)                                =(1/3)*0.01/0.0217                                =0.1536 (3) Machine C: P(Machine C | defect)=P(Machine C and defect)/P(defect)                                =(1/3)*0.05/0.0217                                =0.768

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