Question

 

Over the past 50 years, one-half of all cities that apply have received mass transportation grants from the federal government. In the last round of grants, eight southern cities applied but none received a grant. Assume that all cities were equally qualified to receive the grants. What is the probability that no southern city receives a grant?

 

Answer 1

Solution

Let X = Number of southern cities which received a grant out of eight southern cities that applied.

Then, X ~ B(8, p), where p = probability a southern city received a grant, which is also equal to the proportion of all cities which received the grant.

‘Over the past 50 years, one-half of all cities that apply have received mass transportation grants from the federal government.’ => p = ½ . Thus, X ~ B(8, ½) ………………………………………………………………………………. (1)

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 …………………………………................................…...(2)

[The above probability can also be directly obtained using Excel Function: Statistical, BINOMDIST …........…….(2a)

Now to work out the solution,

Probability that no southern city receives a grant

= P(X = 0)

= (⁸C₀)(½)⁰(½)⁸ [vide (1) and (2)]

= (½)⁸

= 0.0039 Answer

DONE

[Going beyond, vide (2a)

In Excel Function: Statistical, BINOMDIST, entering

Number_s 0

Trials 8

Probability 0.5

Cumulative FALSE, result is 0.0039]