The probability that a call received by a certain switchboard will be a wrong number is 0.02. Use the Poisson distribution to approximate the probability that among 150 calls received by the switchboard, there are at least two wrong numbers. Round your answer to four
Decimal places.
A) 0.8009
B) 0.7760
C) 0.2240
D) 0.1991
E) 0.5768

Question

Quizplus · Accepted Answer

The Poisson distribution can be used to approximate the probability of a given number of events happening in a fixed interval of time or space if these events happen with a known constant rate and independently of the time since the last event. The probability of exactly $k$ events in an interval is given by the formula $P(k) = \frac{\lambda^k e^{-\lambda}}{k!}$, where $\lambda$ is the average rate (mean) of events per interval, and $e$ is the base of the natural logarithm (approximately equal to 2.71828).Given that the probability of a call being a wrong number is 0.02, and there are 150 calls, the expected number of wrong numbers ($\lambda$) is $150 \times 0.02 = 3$.To find the probability of at least two wrong numbers, it's easier to calculate the complement, which is the probability of 0 or 1 wrong number, and subtract it from 1.$$P(0) = \frac{3^0 e^{-3}}{0!} = e^{-3}$$$$P(1) = \frac{3^1 e^{-3}}{1!} = 3e^{-3}$$The probability of 0 or 1 wrong numbers is $P(0) + P(1) = e^{-3} + 3e^{-3} = 4e^{-3}$.The probability of at least two wrong numbers is $1 - (P(0) + P(1)) = 1 - 4e^{-3}$.Calculating this gives a value close to 0.8009, making choice A the correct answer.

The Probability That a Call Received by a Certain Switchboard