A manufacturer produces the same kind of computer chip in two plants. In the first plant, the ratio of defective
chips to good chips is 15:100 and in the second plant, that ratio is 12:100. A buyer of a large number of
chips is aware that some come from the first plant and some from the second. However, she is not aware of how many come from each. The buyer would like to
know the ratio of defective chips to good chips in any given order. Can she determine that ratio? If so, explain
how. If not, explain why not.
Yes, the buyer can determine the ratio of defective chips to good chips in any given order.
To determine the ratio, the buyer needs to calculate the weighted average of the two individual ratios based on the proportion of chips coming from each plant. Let's denote the proportion of chips coming from the first plant as "p" (0 ≤ p ≤ 1) and from the second plant as "1 - p".
The overall ratio of defective chips to good chips can be calculated as follows:
Defective Ratio = (p * First Plant Defective Ratio) + ((1 - p) * Second Plant Defective Ratio)
Given the information provided, the ratio of defective chips to good chips in the first plant is 15:100, and in the second plant, it is 12:100. Substituting these values into the equation, we get:
Defective Ratio = (p * 15:100) + ((1 - p) * 12:100)
Simplifying this equation:
Defective Ratio = (15p + 12(1 - p))/100
Defective Ratio = (15p + 12 - 12p)/100
Defective Ratio = (3p + 12)/100
Therefore, the buyer can determine the ratio of defective chips to good chips in any given order by using the equation Defective Ratio = (3p + 12)/100, where "p" represents the proportion of chips coming from the first plant.