A survey of 100 companies shows that 75 of them have installed wireless local area networks (WLANs) on their premises. If three of these companies are chosen at

Question

A survey of 100 companies shows that 75 of them have installed wireless
local area networks (WLANs) on their premises. If three of these
companies are chosen at random without replacement, what is the
probability that each of the three has installed WLANs?

Answer 1

This is sampling without replacement.  Thus, it follows the hypergeometric distribution.

P(3) = C(75,3)C(25,0)/C(100,3) - we are selecting 3 from 75 and none from the remaining 25.

This equals

(75*74*73)/3!*1/((100*99*98)/3!)) = 75*74*73/(100*99*98) = 37*73/(2*33*98) = 2701/6498 = .4176

 

(Note that (3/4)^3 = .4219; sampling without replacement reduces the probability of all successes or all failures)