Question

About 58% of students go to a college within 100 miles of their home. If you choose a random sample of 10 students, what is the probability that at least four students go to a college within 100 miles of their home?

Answer 1

Solution

Let X = Number of students who go to a college within 100 miles of their home. Then, X ~ B(n, p), where

n = sample size = 10 [given] and

p = probability a randomly chosen students goes to a college within 100 miles of home = 0.58 [given]

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}, x = 0, 1, 2, ……. , n …………….......................................……..(1)

[The above probability can also be directly obtained using Excel Function of Binomial Distribution: BINOMDIST(Number_s:Trials:Probability_s:Cumulative), what is within brackets is (x:n:p:True)] ….(1a)

Now to work out the solution,

Probability that at least four students go to a college within 100 miles of their home

= P(X ≥ 4)

= Σ(x = 4 to 10){ (¹⁰C_x)(0.58^x)(0.42)^{10 – x} } [vide (1)]

= 1 – 0.071164 [vide (1a)]

= 0.928836 Answer