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 [ a new business will survive for 2 years ] = p [ surviving time > 2 ] = e^(- 2 / 3 ) = e^(-2/3)...
b) p [ a new business will fail in its second year ] = p [ surviving time is in between 1 and 2 year ]

= ( 1 - e^(-2/3) ) - ( 1 - e^(-1/3) ) = e^(-1/3) - e^(-2/3) = e^(-1/3) [ 1 - e^(-1/3) ] .......

c) probability that at least one will survive two years = PROB. THAT NO ONE WILL SURVIVE FOR 2 YEARS
= ( 5 C 0) (1/3)^0 ( 2/3) ^ 5 = 0.1316872...

Assuming, x = the no. of years one will survive..

prob. of success = 1/3..and prob. of success is independent for each person...