Laidoff workers who become entrepreneurs because they cannot

Laid-off workers who become entrepreneurs because they cannot find meaningful employment with another company are known as entrepreneurs by necessity. A major national newspaper reports that these entrepreneurs by necessity are less likely to grow into large business than are entrepreneurs by choice. This article states that 74% of entrepreneurs in a certain area are entrepreneurs by choice and 26% are entrepreneurs by necessity. Only 1% of entrepreneurs by necessity expect their new business to employ 20 or more people within five years, whereas 11% of entrepreneurs by choice expect to employ at least 20 people within five years.

If an entrepreneur is selected at random and that individual expects that his or her new business will employ 20 or more people within five years, what is the probability that this individual is an entrepreneur by choice?

Solution

Let

E = will employ 20 or more in 5 years
N = by necessity
C = by choice

Then

P(C|E) = P(C) P(E|C) / P(E)

As

P(E) = P(C) P(E|C) + P(N) P(E|N) = 0.74*0.11 + 0.26*0.01 = 0.084

Thus,

P(C|E) = P(C) P(E|C) / P(E) = 0.74*0.11/0.084 = 0.969047619 [ANSWER]