Question

the communications monitoring company postini has reported that 92% of e - mail messages are spam. you randomly chose 10 e - mails. what is the probability you get exactly 8 spam messages? (round your answer to 4 decimal places)

Explanation:

Step1: Identify the distribution

This is a binomial probability problem. The binomial probability formula is $P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k}$, where $n$ is the number of trials, $k$ is the number of successes, $p$ is the probability of success on a single trial. Here, $n = 10$, $k = 8$, $p = 0.92$.

Step2: Calculate the combination

First, calculate $\binom{10}{8}$. The combination formula is $\binom{n}{k} = \frac{n!}{k!(n - k)!}$. So $\binom{10}{8} = \frac{10!}{8!(10 - 8)!} = \frac{10!}{8!2!} = \frac{10\times9}{2\times1} = 45$.

Step3: Calculate $p^k$ and $(1 - p)^{n - k}$

$p^k = 0.92^8$, and $(1 - p)^{n - k} = (1 - 0.92)^{10 - 8} = 0.08^2$.

Step4: Multiply the components

Now, multiply these together: $P(X = 8) = 45\times0.92^8\times0.08^2$.

Calculate $0.92^8 \approx 0.51321887$, $0.08^2 = 0.0064$. Then $45\times0.51321887\times0.0064 \approx 45\times0.0032846008 \approx 0.147807036$.

Answer:

0.1478