Question

The student body of a large university consists of 30% Business majors. A random sample of 6 students is selected.

A. What is the probability that exactly 4 are business majors?

B. What is the probability that no more than 2 are business majors?

C. What is the probability that at least 3 are business majors?

D. What is the probability that less than 5 are business majors?

Answer 1

Solution

Back-up Theory

If X ~ B(n, p). i.e., X has Binomial Distribution with parameters n and p, where

n = number of trials and p = probability of one success, then, probability mass function (pmf)

of X is given by p(x) = P(X = x) = (ⁿC_x)(p^x)(1 - p)^{n – x},...............................................................................………..(1)

[This probability can also be directly obtained using Excel Function: Statistical, BINOMDIST]........................... (1a)

Now to work out the solution,

Let X = number of students is a sample of 6 who are business majors. Then, X ~ B(6, 0.3) [0.3 = 30%] ............ (2)

Part (A)

Probability that exactly 4 are business majors

= P(X = 4)

= 0.0595 [vide (1a) and (2)] Answer 1

Part (B)

Probability that no more than 2 are business majors

= P(X ≤ 2)

= 0.7443 [vide (1a) and (2)] Answer 2

Part (C)

Probability that at least 3 are business majors

= P(X ≥ 3)

= 1 - P(X ≤ 2)

= 0.2557 [vide (1a) and (2)] Answer 3

Part (D)

Probability that less than 5 are business majors

= P(X < 5)

= P(X ≤ 4)

= 0.9891 [vide (1a) and (2)] Answer 2

DONE