What is the probability that the business makes a profit in exactly two of its first three years?

Question

Assume that the business makes a profit with probability 0.2 in the first year. For each year thereafter, the business makes a profit with probability 0.7 if it made a profit in the previous year, and with probability 0.2 if it did not make a profit in the previous year.
What is the probability that the business makes a profit in exactly two of its first three years?

Answer 1

Let W (for Win) denote profit in each year and L denote loss.  We are given 

 

P(W) = 0.2 in the first year so P(L) = 0.8 in the first year 

 

If there was a profit in the first year then we have 

P(WW) = 0.2*0.7 = 0.14 = probability of profit in both years

P(WL) = 0.2*0.3 = 0.06 = probability of profit in year 1 and loss in year 2

P(LW) = 0.8*0.2 = 0.16 = probability of loss in year 1 and profit in year 2

P(LL) = 0.8*0.8 = 0.64 probability of loss in both years 

 

Continuing to the third year we have 

 

P(WWW) = 0.14*0.7 = 0.098

P(WWL) = 0.14*0.3 = 0.042

P(WLW) = 0.06*0.2 = 0.012

P(WLL) = 0.06*0.8 = 0.048

P(LWW) = 0.16*0.7 = 0.112

P(LWL) = 0.16*0.3 = 0.048

P(LLW) = 0.64*0.2 = 0.128

P(LLL) = 0.64*0.8 = 0.512 

 

Of these 8 possibilities, 3 have a profit in exactly 2 of the 3 years, they are 

 

P(WWL) + P(WLW) + P(LWW) = 0.042 + 0.012 + 0.112 = 0.166