Assume that the business makes a profit with probability

0.8 in the first year. For each year thereafter, the business makes a profit
with probability 0.8 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?

To find the probability that the business makes a profit in exactly two of its first three years, we need to consider all the possible combinations of profits and losses in those three years.

Let's break it down step by step:

Step 1: List all possible combinations
The business can make a profit (P) or not make a profit (NP) in any given year. For three years, we have 2^3 = 8 possible combinations:
1. PPP
2. PPN
3. PNP
4. PNN
5. NPP
6. NPN
7. NNP
8. NNN

Step 2: Assign probabilities to each combination
We know that the probability of making a profit in the first year is 0.8, and the probability of not making a profit is 0.2. For subsequent years, the probability of making a profit is determined by whether the previous year was profitable or not.

Using this information, we can assign the probabilities to each combination:
1. PPP: 0.8 * 0.8 * 0.8 = 0.512
2. PPN: 0.8 * 0.8 * 0.2 = 0.128
3. PNP: 0.8 * 0.2 * 0.8 = 0.128
4. PNN: 0.8 * 0.2 * 0.2 = 0.032
5. NPP: 0.2 * 0.8 * 0.8 = 0.128
6. NPN: 0.2 * 0.8 * 0.2 = 0.032
7. NNP: 0.2 * 0.2 * 0.8 = 0.032
8. NNN: 0.2 * 0.2 * 0.2 = 0.008

Step 3: Calculate the probability of making a profit in exactly two years
To find the desired probability, we need to sum up the probabilities of the combinations where the business made a profit in exactly two years.

From the list, the combinations that meet this criteria are PPN, PNP, and NPP. Their probabilities are 0.128, 0.128, and 0.128.

Summing these probabilities, we get:
0.128 + 0.128 + 0.128 = 0.384

Therefore, the probability that the business makes a profit in exactly two of its first three years is 0.384 or 38.4%.

To calculate the probability that the business makes a profit in exactly two of its first three years, we can use the binomial probability formula:

P(X = k) = (nCk) * p^k * (1-p)^(n-k)

where:
- P(X = k) is the probability that X, the number of successes, equals k
- n is the total number of trials
- p is the probability of success in a single trial
- (nCk) is the number of combinations of n items taken k at a time
- k is the number of successful trials

In this case, we want to find the probability that the business makes a profit in exactly two out of three years. Therefore, we substitute the following values into the formula:

n = 3 (the total number of years)
p = 0.8 (the probability of making a profit in a given year)
k = 2 (the number of years the business makes a profit)

Using the formula, we can calculate the probability as follows:

P(X = 2) = (3C2) * 0.8^2 * (1-0.8)^(3-2)

P(X = 2) = (3C2) * 0.8^2 * 0.2^1

To calculate (3C2), which is the number of combinations of 3 items taken 2 at a time, we use the formula:

(3C2) = 3! / (2! * (3-2)!)

(3C2) = 3! / (2! * 1!)

(3C2) = 3

Substituting this value into the original formula:

P(X = 2) = 3 * 0.8^2 * 0.2^1

P(X = 2) = 3 * 0.64 * 0.2

P(X = 2) = 0.384

Therefore, the probability that the business makes a profit in exactly two of its first three years is 0.384 or 38.4%.