The binomial probability for P(1

Question

The binomial probability for P(1 < x < 3), where n = 30, p = .01 is .2601.If you use the Poisson distribution to approximate this binomial probability, what probability would you produce?
A).1775
B).1413
C).2588
D).2263
E).1917

Quizplus · Accepted Answer

The Poisson distribution can be used to approximate the binomial distribution when n is large and p is small. The mean ($\lambda$) of the Poisson distribution is $n \times p = 30 \times 0.01 = 0.3$. The probability of getting 2 successes (since we want $1 < x < 3$, which means x=2) in a Poisson distribution is calculated using the formula $P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}$. Plugging in the values gives $P(X=2) = \frac{e^{-0.3} \times 0.3^2}{2!} = \frac{e^{-0.3} \times 0.09}{2} = 0.2588$, which is a close approximation to the binomial probability given.

The Binomial Probability for P(1 < X < 3), Where