Question

Assume that the business makes a profit with probability 0.3 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

make a tree where end endpoint has two branches coming off it
Call one branch P for profit, the other branch NP for non-profit.
The first year has 2 endpoints, the 2nd year has 4 endpoints, and the 3rd year has 8 endpoints.

Place the appropriate P or NP at the end of each branch, and write the probability on each branch

prob of profit in 1st year = .3

pathways to get to P in year two
= (.7)(.3) + (.7)(.2) = .35
pathways to get to P in year 3
= (.7^2)(.3) + (.3^2)(.2) + (.7^2)(.2) + (.7)(.8)(.2) = .375

so to have 2 out of the 3 years it could be years
1,2 or 1,3 or 2,3

prob = (.3)(.35) + (.3)(.375) + (.35)(.375) = .34875

Check my calculations, this was a tricky question.
I did all up all the P and NP prob in year 3 and they added to 1 as they should