A company manufactures chairs at a cost of $50 each.

Question

The following are probabilities of defective chairs in a given production run: probability of 1% defective chairs: 40%; probability of 2% defective chairs: 35%; probability of 3% defective chairs: 25%. When a defective chair is detected, it must be removed from the assembly line and replaced. This process adds an extra $10 to the cost of the replaced chair. What is the expected cost to the company of a batch of 500 chairs?

Answer 1

Think of  this problem in the following way. Suppose that each day you build 500 chairs and each day you keep track of the number of defective chairs. After 100 days you tally up your results and you find that 40% of the time only 5 chairs were defective (1%), 35% of the time 10 chairs were defective (2%) and finally 25% of the time 15 chairs were defective (3%). Evidently there were no days where all the chairs came out in good shape and there were no days where 3 chairs were defective or 7 chairs were defective. In this stylized problem either 5, 10, or 15 chairs were defective. Note that the probabilities add up to 100% so there are no other categories of defects.

 

Let's consider the cost when 5 chairs were defective. This would mean that you would have to make 505 chairs. You threw away 5 but they cost $50 to make. Then you had to make 5 more at a cost of $60. The total cost of the chairs made would be 500*50 + 5*60, or $125,300.

 

Now, what is the cost if 10 chairs are defective? That would be 500*50 + 10*60, or $125,600.

 

Similarly the cost with 15 defects is $125,900.

 

Now, all we have to do is multiply these by their respective probabilities, and the answer is $125,555 if I did my math right.