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 0.3 in the first year. For each year thereafter, the business makes a profit with probability 0.6 if it made a profit in the previous year, and with probability 0.3 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

Because of the complexity of the various probabilities, I think you have to look at each of the possible cases and figure out the probability for each one.

 

There are three ways this can happen:

 

1. Profitable in years 1 and 2, and not year 3

2. Profitable in years 1 and 3, and not year 2

3. Profitable in years 2 and 3, and not year 1

 

Each of these is mutually exclusive -- you can't have more than one of these cases occur.  So when you calculate the probabilities of each one, you just have to add them together to get the probability that one of them will occur.

 

Case 1:  The probability of being profitable in year 1 is 0.3.  If it's profitable in year 1, then the probability of being profitable in year 2 is 0.6.  If it's profitable in year 2, the probability that it will be profitable in year 3 is 0.6, so the probability that it WON'T be profitable in year 3 is 1 - 0.6, or 0.4.  So the probability that this case will occur is 0.3 * 0.6 * 0.4 = 0.072

 

Case 2:  The probability of being profitable in year 1 is 0.3.  If it's profitable in year 1, then the probability of being profitable in year 2 is 0.6, so the probability that it WON'T be profitable in year 2 is 1 - 0.6, or 0.4. If it's NOT profitable in year 2, then the probability that it will be profitable in year 3 is 0.3.  So the probability of this case occurring is 0.3 * 0.4 * 0.3 = 0.036

 

Case 3:  The probability of being profitable in year 1 is 0.3, so the probability of NOT being profitable in year 1 is 0.7.  If it's not profitable in year 1, then the probability of being profitable in year 2 is 0.3.  If it's profitable in year 2, then the probability of being profitable in year 3 is 0.6.  So the probability of this case occurring is 0.7 * 0.3  * 0.6 = 0.126

 

The sum of the three probabilities is 0.072 + 0.036 + 0.126 = 0.234