Question

1. 20% of all restaurants become successful chains. if 10 new restaurants just opened up near chs, then what is the probability that exactly 7 of those restaurants become successful chains?

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, $n = 10$, $k = 7$, $p = 0.2$.

Step2: Calculate the combination

First, calculate $\binom{10}{7}$. The formula for combinations is $\binom{n}{k} = \frac{n!}{k!(n - k)!}$. So $\binom{10}{7} = \frac{10!}{7!(10 - 7)!} = \frac{10!}{7!3!} = \frac{10\times9\times8}{3\times2\times1} = 120$.

Step3: Calculate the probability terms

Next, calculate $p^k = 0.2^7$ and $(1 - p)^{n - k} = 0.8^{3}$. $0.2^7 = 1.28\times10^{-5}$, $0.8^3 = 0.512$.

Step4: Multiply the terms

Now, multiply these together: $P(X = 7) = 120\times1.28\times10^{-5}\times0.512$.
First, $120\times1.28\times10^{-5}=1.536\times10^{-3}$, then $1.536\times10^{-3}\times0.512 = 7.86432\times10^{-4}$.

Answer:

The probability is approximately $7.86\times10^{-4}$ (or $0.000786$).