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?
Well, let's get down to business! The probability that the business makes a profit in exactly two of its first three years can be calculated using some clown math.
To solve this problem, let's break it down into different scenarios.
Scenario 1: The business makes a profit in the first year, then in the second year, but not in the third year. The probability for this scenario is 0.3 (first year profit) * 0.7 (second year profit) * 0.2 (third year no profit) = 0.042.
Scenario 2: The business makes a profit in the first year and third year, but not in the second year. The probability for this scenario is 0.3 (first year profit) * 0.2 (second year no profit) * 0.7 (third year profit) = 0.042.
Scenario 3: The business does not make a profit in the first year, but makes a profit in the second and third year. The probability for this scenario is 0.7 (first year no profit) * 0.7 (second year profit) * 0.7 (third year profit) = 0.343.
Adding up the probabilities from all these scenarios, we get 0.042 + 0.042 + 0.343 = 0.427.
So, the probability that the business makes a profit in exactly two of its first three years is approximately 0.427. Keep in mind that this is just one possibility, and there may be other scenarios as well.
Remember, in clown math, probabilities are always funny business!
To calculate the probability that the business makes a profit in exactly two of its first three years, we can consider the different scenarios.
First, let's define some variables:
P1 = probability of making a profit in the first year = 0.3
P2 = probability of making a profit in the second year given a profit in the previous year = 0.7
P3 = probability of making a profit in the third year given a profit in the previous year = 0.7
P4 = probability of making a profit in the second year given no profit in the previous year = 0.2
P5 = probability of making a profit in the third year given no profit in the previous year = 0.2
Now, let's calculate the probability for each scenario:
1. Profit in the first year, profit in the second year, no profit in the third year:
P1 * P2 * P5 = 0.3 * 0.7 * 0.2 = 0.042
2. Profit in the first year, no profit in the second year, profit in the third year:
P1 * P4 * P3 = 0.3 * 0.2 * 0.7 = 0.042
3. No profit in the first year, profit in the second year, profit in the third year:
(1 - P1) * P2 * P3 = 0.7 * 0.7 * 0.7 = 0.343
Now, add up the probabilities from each scenario to get the overall probability:
0.042 + 0.042 + 0.343 = 0.427
Therefore, the probability that the business makes a profit in exactly two of its first three years is 0.427 or 42.7%.
To find the probability that the business makes a profit in exactly two of its first three years, we can use the concept of probability calculation and the given information.
First, let's break down the problem step by step:
1. We have three years in total, and we need to determine the probability of making a profit in exactly two of those years.
2. We need to consider all possible combinations of profit and loss in these three years, i.e., PPL, PLP, LPP, where P denotes a profit and L denotes a loss.
3. From the given information, we know the probabilities of making a profit or not in each subsequent year based on the previous year's outcome.
Using this information, let's calculate the probability for each of the three combinations mentioned above and add those probabilities together to find the desired result:
1. PPL: In the first year, the probability of making a profit is 0.3. In the second year, the probability of making a profit after a profit is 0.7, and in the third year, the probability of making a profit after a loss is 0.2. Therefore, the probability of the PPL combination is: 0.3 * 0.7 * 0.2.
2. PLP: In the first year, the probability of making a profit is 0.3. In the second year, the probability of making a profit after a loss is 0.2, and in the third year, the probability of making a profit after a profit is 0.7. Therefore, the probability of the PLP combination is: 0.3 * 0.2 * 0.7.
3. LPP: In the first year, the probability of making a profit is 0.2. In the second year, the probability of making a profit after a profit is 0.7, and in the third year, the probability of making a profit after a profit is 0.7. Therefore, the probability of the LPP combination is: 0.2 * 0.7 * 0.7.
Finally, we can add all these probabilities together to get the probability of making a profit in exactly two of the first three years:
Probability = (0.3 * 0.7 * 0.2) + (0.3 * 0.2 * 0.7) + (0.2 * 0.7 * 0.7) = 0.042 + 0.042 + 0.098 = 0.182
Therefore, the probability that the business makes a profit in exactly two of its first three years is 0.182, or 18.2%.
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 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