We have a group of N=14 government officials. In this group there ar 4 spies, what means a proportion of spies of p=4/14=2/7.
We have to calculate the probability of picking a group or sample of 4 government officials (n=4) and none of them is a spy.
We can calculate this probability with the binomial distribution: the probability of "success" is p=2/7 (the proportion of spies in the government officials) and the sample size is n=4, the size of the group.
As we define success as picking a spy, we will calculate the probability of "failure" as the probability of having no spy in the group: this means calculate P(x=0), with x meaning the number of spies.
This can be done as:
[tex]\begin{gathered} P(x=k)=\binom{n}{k}p^k(1-p)^{n-k} \\ P(x=0)=\binom{4}{0}(\frac{2}{7})^0(1-\frac{2}{7})^4 \\ P(x=0)=\frac{4!}{0!4!}\cdot1\cdot(\frac{5}{7})^4 \\ P(x=0)=1\cdot1\cdot(\frac{5}{7})^4 \\ P(x=0)\approx0.26 \end{gathered}[/tex]Then, the probability of selecting no spy in a sample of 4 government agents is P=0.26.
Answer: the probability is 0.26.
