Question

Thirty-two percent of the students in a management class are graduate students. A random sample of 5 students is selected. Using the binomial probability function, determine the probability that the sample contains fewer than two graduate students? (Please express answer to four decimal places in the following form: 2.5555 or 2.0001).

Answer 1

0.4875

Explanation:

For each student, there are only two possible outcomes. Either they are a graduate student, or they are not. The probability of a student being a graduate student is independent from other students. So we use the binomial probability distribution to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

In which
`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

And p is the probability of X happening.

Thirty-two percent of the students in a management class are graduate students.

This means that
`p = 0.32`

A random sample of 5 students is selected.

This means that
`n = 5`

Determine the probability that the sample contains fewer than two graduate students?

`P(X < 2) = P(X = 0) + P(X = 1)`

In which

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

`P(X = 0) = C_(5,0).(0.32)^(0).(0.68)^(5) = 0.1454`

`P(X = 1) = C_(5,1).(0.32)^(1).(0.68)^(4) = 0.3421`

`P(X < 2) = P(X = 0) + P(X = 1) = 0.1454 + 0.3421 = 0.4875`

0.4875 = 48.75% probability that the sample contains fewer than two graduate students