The probability that a call received by a certain switchboard will be a wrong number is 0.02.Use the Poisson approximation to the binomial distribution to find the probability that among 140 calls received by the switchboard,there are at least two wrong numbers.
A)0.5305
B)0.7616
C)0.7689
D)0.2384
E)0.2311

Question

Quizplus · Accepted Answer

Let X be the number of wrong numbers among 140 calls. We want to find P(X ≥ 2).

Since n = 140 is large and p = 0.02 is small, we can use the Poisson approximation to the binomial distribution:

X ~ Poisson(λ), where λ = np = 140 × 0.02 = 2.8.

Then,

P(X ≥ 2) = 1 - P(X = 0) - P(X = 1)
             = 1 - e^(-λ) - λe^(-λ)
             = 1 - e^(-2.8) - 2.8e^(-2.8)
             ≈ 0.7689

Therefore, the best choice is (C).

The Probability That a Call Received by a Certain Switchboard