Answer to Question #152351 in Statistics and Probability for Yosef

Question

A plane can accommodate 20 people; statistics show that 25% of customers who have booked do not come. Let X be the random variable: "number of customers who come after reservation".

1. What is the distribution of X? 2. What is the probability so that X is equal to 15?

Answer 1

1. The random variable $X$ has the binomial distribution: $X\sim Bin(n, p)$

$$p(X=x)=\dbinom{n}{x}p^x(1-p)^{n-x}$$

Given $p=0.25, n=20$

$X\sim Bin(20, 0.25)$

$$p(X=x)=\dbinom{20}{x}(0.25)^x(1-0.25)^{20-x}$$

2.

$$p(X=15)=\dbinom{20}{15}(0.25)^{15}(1-0.25)^{20-15}$$

$$=\dfrac{20!}{5!(20-15)!}(0.25)^{15}(0.75)^{5}$$

$$\approx0.0000034265$$

The probability so that X is equal to 15 is approximately $3.4265\times10^{-6}$