Question

In a large university, statistics show that 75% of students live in dormitories. Thus, for any student chosen at random, the university administration estimates a probability of 0.75 that the student will live in dormitories. A random sample of 5 students is selected. Answer the following questions.

1.What is the probability that the sample contains exactly 3 students who live in the dormitories?

2.What is the probability that fewer than 2 students live in the dormitories?

3.What is the expected number and variance of students who live in the dormitories when 5 students are randomly selected?

Answer 1

1)

Here, n = 5, p = 0.75, (1 - p) = 0.25 and x = 3
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X = 3)
P(X = 3) = 5C3 * 0.75^3 * 0.25^2
P(X = 3) = 0.2637

b)

Here, n = 5, p = 0.75, (1 - p) = 0.25 and x = 2
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X < 2).
P(X < 2) = (5C0 * 0.75^0 * 0.25^5) + (5C1 * 0.75^1 * 0.25^4)
P(X < 2) = 0.001 + 0.0146
P(X < 2) = 0.0156

c)

mean = np = 5 * 0.75 = 3.75

varaince = npq

= 5 * 0.75 *(1-0.75)

= 0.9375