4 Suppose that onethird of new businesses fail in one year O

4. Suppose that one-third of new businesses fail in one year. Of those remaining after one year, one-third fail in their second year of operation.

(a) What is the probability that a new business will survive for two years?

(b) what is the probability that a new business will fail in its second year?

(c) Out of a group of five new businesses, what is the probability that at least one will survive two years? Upon what assumption is your answer based?

Solution

a)

P(survive two years) = P(survive first year) P(survive second|survive first)

= (1-1/3)*(1-1/3)

= 4/9 or 0.444444444 [ANSWER]

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b)

P(fail on second year) = P(survive first year) P(fail second year|survive first)

= (1-1/3)*(1/3)

= 2/9 or 0.222222222 [ANSWER]

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c)

P(at least one survives 2 years) = 1 - P(none survives 2 years)

As

P(not survive 2 years) = P(fail first year) + P(fail 2nd year)

P(not survive 2 years) = 1/3 + 2/9 = 5/9

Thus,

P(none survives out of 5) = (5/9)^5 = 0.052922149

Thus,

P(at least one survives out of 5) = 1 - 0.052922149 = 0.947077851 [ANSWER]