The production department has installed a new spray machine to paint automobile doors. As is common with most spray guns,unsightly blemishes often appear because of improper mixtures or other problems. A worker counted the number of blemishes on each door. Most doors had no blemishes; a few had one; a very few had two; and so on. The average number was 0.5 per door. The distribution of blemishes followed the Poisson distribution. Out of 10,000 doors painted,about how many would have no blemishes?
A)About 6,065
B)About 3,935
C)About 5,000
D)About 500

Question

Quizplus · Accepted Answer

Given that the distribution of blemishes follows the Poisson distribution and the average number of blemishes per door is 0.5. We know that the Poisson distribution with parameter λ gives the count of events occurring in a fixed interval of time or space, where events occur randomly and independently with an average rate of λ. Here, a blemish on the door is the event, and the Poisson parameter λ = 0.5.
The probability of observing exactly x blemishes on a single door is given by the Poisson probability mass function P(x) = (e^-λ * λ^x) / x!
The probability of observing 0 blemishes on a single door is P(0) = (e^-0.5 * 0.5^0) / 0! = e^-0.5 = 0.6065 (approximately)
Therefore, the proportion of doors with no blemishes is 0.6065. Out of 10,000 doors painted, the number of doors with no blemishes would be approximately 0.6065 * 10,000 = 6,065. Hence, the best choice is A.

The Production Department Has Installed a New Spray Machine to Paint