The machining center at a production facility has three lathes of differing ages and thus precision. The oldest, machine A, produces finished units of product of which 88% are good, 8% are blemished, and 4% unusable. Machine B produces 92% good, 6% blemished, and 2% unusable. The newest machine, machine C, turns out product that is 96% good, 3% blemished, and 1% unusable. If machine A produces ¼ of the company’s output and machine B turns out 1/3 of the output, what fraction of the company’s product is good? What percentage is blemished? If an unit of product is selected at random and found to be blemished, what is the probability that it was produced on machine B?
To find the fraction of the company's product that is good, we need to calculate the weighted average of the percentages of good units from each machine, based on their respective proportions of output.
Let's start by finding the proportion of output from each machine:
- Machine A produces 1/4 (or 25%) of the company's output.
- Machine B produces 1/3 (or 33.33%) of the company's output.
- Machine C produces the remaining fraction, which is 1 - (1/4 + 1/3) = 5/12 (or approximately 41.67%) of the company's output.
Now, let's calculate the weighted average of the percentages of good units produced by each machine:
- For machine A: 88% good * 25% output = 22% contribution to the overall output.
- For machine B: 92% good * 33.33% output = 30.66% contribution to the overall output.
- For machine C: 96% good * 41.67% output = 40% contribution to the overall output.
To find the fraction of the company's product that is good, we sum up the contributions from each machine: 22% + 30.66% + 40% = 92.66%.
Therefore, approximately 92.66% of the company's product is good.
To find the percentage that is blemished, you need to calculate the weighted average of the percentages of blemished units from each machine using the same methodology as above.
- For machine A: 8% blemished * 25% output = 2% contribution to the overall output.
- For machine B: 6% blemished * 33.33% output = 1.99% contribution to the overall output.
- For machine C: 3% blemished * 41.67% output = 1.25% contribution to the overall output.
Adding up these contributions, we find that approximately 5.24% of the company's product is blemished.
Now, to calculate the probability that a blemished unit was produced on machine B, we need to determine the proportion of blemished units that came from machine B.
The total proportion of blemished units is the sum of the contributions from each machine:
- For machine A: 2% contribution
- For machine B: 1.99% contribution
- For machine C: 1.25% contribution
The proportion of blemished units produced by machine B is 1.99% out of the total 5.24%, which gives us a probability of approximately 1.99% / 5.24% = 0.379 or 37.9%.
Therefore, given that a unit is blemished, the probability that it was produced on machine B is approximately 37.9%.