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 0.5 in the first year. For each year thereafter, the business makes a profit with probability 0.9 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

This is the type of problem that I like to use a tree diagram to do. However, the memory allocation for these answers is limited so I can't show you what it would look like. Instead I'll show the possible outcomes and their associated probabilities:

In the outcomes below, "P" means they make a profit, "N" means they do NOT make a profit and the three values represent year 1, year 2, and year 3. I've put an asterisk on the outcomes that have EXACTLY 2 years of profit.

PPP

PPN*

PNP*

PNN

NPP*

NPN

NNP

NNN

Let's look at those that have exactly 2 years of profit in more detail:

For PPN, the probability is (0,5)(0.9)(0.1) = 0.045

For PNP, the probability is (0,5)(0.1)(0.2) = 0.010

For NPP, the probability is (0,5)(0.2)(0.9) = 0.090

So the probability that any of these could happen is their sum: 0.045 + 0.010 + 0.090 = 0.145