Question

A survey unofficially claimed that in every five young executives, only one practices good reading habits.

a. What is the probability that out of 10 young executives, two executives practice good reading habits?

b. What is the probability that at least five out of 20 young executives practice good reading habits?

Answer 1

Let $X=$ the number of executives with good reading habits: $X\sim Bin (n, p)$

Given $p=1/5=0.2, q=1-p=0.8$

a.

$n=10$

$$P(X=2)=\dbinom{10}{2}(0.2)^{2}(0.8)^{10-2}=0.301989888$$

b.

$n=20$

$$P(X\ge5)=1-P(X=0)-P(X=1)$$

$$-P(X=2)-P(X=3)-P(X=4)$$

$$1-\dbinom{20}{0}(0.2)^{0}(0.8)^{20-0}-\dbinom{20}{1}(0.2)^{1}(0.8)^{20-1}$$

$$-\dbinom{20}{2}(0.2)^{2}(0.8)^{20-2}-\dbinom{20}{3}(0.2)^{3}(0.8)^{20-3}$$

$$-\dbinom{20}{4}(0.2)^{4}(0.8)^{20-4}=0.3703517361$$