Answer to Question #263799 in Statistics and Probability for Bravo

Question

A management of a factory claims that 70% of all their employees have been vaccinated. If a sample of 15 people are chosen, find probability that more than 13 employees actually get vaccinated.

Answer 1

Let $X=$ the number of employees who have been vaccinated: $X\sim Bin(n, p).$

Given $n=15, p=0.7, q=1-p=1-0.7=0.3.$

$$P(X>13)=P(X=14)+P(X=15)$$

$$=\dbinom{15}{14}(0.7)^{14}(0.3)^{15-14}+\dbinom{15}{15}(0.7)^{15}(0.3)^{15-15}$$

$$=0.030520038278205+0.004747561509943$$

$$=0.035267599788148$$

$$\approx0.0352676$$

The probability that more than 13 employees actually get vaccinated is $0.0352676.$