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 formula is $P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}$, where $\lambda$ is the mean number of successes in the given time interval or spatial area. Here, $\lambda = 150 \times 0.02 = 3$. To find the probability of at least two wrong numbers, we calculate the complement of the probability of having 0 or 1 wrong numbers. Using the Poisson formula:- For 0 wrong numbers: $P(X=0) = \frac{e^{-3} 3^0}{0!} = e^{-3}$- For 1 wrong number: $P(X=1) = \frac{e^{-3} 3^1}{1!} = 3e^{-3}$The probability of having 0 or 1 wrong numbers is $e^{-3} + 3e^{-3} = 4e^{-3}$. The complement, which is the probability of having at least 2 wrong numbers, is $1 - 4e^{-3}$. Calculating this gives a value close to 0.8009, hence the correct answer is A.

The Probability That a Call Received by a Certain Switchboard