Question

what is the probability that at least five out of 20 young executives practice good reading habits

Answer 1

Assume that there are two possibilities for each of $20$ young executives: a person practices good reading habits or not. There are $2^{20}$ different possibilities. The number $2^{20}$ is obtained by using the multiplication principle of combinatorics. Denote by $N$ a number of young executives that practice good reading habits. $N$ is a random variable. We assume that chances to practice good reading habits or not for all persons are equal. The task is to find $P(N\geq5)$; $P(N\geq5)=1-P(N<5)=1-\sum_{i=0}^4P(N=i).$

$P(N=i)=\frac{C_{20}^i}{2^{20}};$ $C_{20}^i=\frac{20!}{i!(20-i)!};$ $C_{20}^i$ is a binomial coefficient.

We get: $P(N\geq5)=\frac{2^{20}-C_{20}^0-C_{20}^1-C_{20}^2-C_{20}^3-C_{20}^4}{2^{20}}\approx0.994$.

Thus, the probability that at least five persons practice good reading habits is: $0.994.$