It has been found that 80% of all printers used operate correctly at the time of installation. A seller sells 10 units during a month.
1)find the prob.that at least 9 of the printers operate correctly at the time of installation.
2)Consider 5 months in which 10 units are sold per month. What is the prob.that at least 9 units operate correctly in each of the 5 months
For 1st part :
Let p(o)=probability that a given printers operates correctly
So we have to find p(o>=9) here.
How to apply the value 80% in this one and solve this question?
I guess I was able to sort the question out using binomial distribution
To solve the first part of the question, we can use the binomial probability formula. In this case, we want to find the probability that at least 9 out of 10 printers operate correctly at the time of installation.
1) Finding the probability that exactly 9 printers operate correctly:
Using the binomial probability formula, we have:
P(X = k) = C(n, k) * p^k * q^(n-k)
where n is the number of trials (10 printers), k is the number of successes (9 printers operating correctly), p is the probability of success (80% or 0.8 in your case), q is the probability of failure (1 - p), and C(n, k) is the number of combinations of n items taken k at a time.
P(X = 9) = C(10, 9) * (0.8)^9 * (0.2)^1 = 10 * (0.8)^9 * (0.2)^1
2) Finding the probability that exactly 10 printers operate correctly:
Using the same formula, we have:
P(X = k) = C(n, k) * p^k * q^(n-k)
P(X = 10) = C(10, 10) * (0.8)^10 * (0.2)^0 = 1 * (0.8)^10 * (0.2)^0
3) Finding the probability that at least 9 printers operate correctly:
To find the probability of at least 9 printers operating correctly, we need to sum the probabilities of exactly 9 and 10 printers operating correctly:
P(X >= 9) = P(X = 9) + P(X = 10)
P(X >= 9) = 10 * (0.8)^9 * (0.2)^1 + 1 * (0.8)^10 * (0.2)^0
Simplifying the equation will give you the final answer.
For the second part, where 5 months are considered with 10 units sold each month, you can apply the same logic and calculations for each month, and then multiply the probabilities together to find the probability that at least 9 units operate correctly in all 5 months.
To solve this question, we can use the binomial distribution formula. The formula for the binomial distribution gives us the probability of getting exactly k successes in n independent Bernoulli trials, where the probability of success for each trial is p.
For the first part of the question, we need to find the probability that at least 9 of the 10 printers operate correctly. We know that p = 0.8, which represents the probability that a given printer operates correctly at the time of installation.
To find the probability of at least 9 printers operating correctly, we need to calculate the probability of 9, 10 printers operating correctly and sum them up.
To calculate the probability of exactly k successes in n trials using the binomial distribution formula, we use the following formula:
P(X=k) = (nCk) * p^k * (1-p)^(n-k)
Where:
- nCk is the number of combinations, calculated as n! / (k!(n-k)!).
- p is the probability of success on a single trial.
- k is the number of successes.
For the first part of the question:
- We have n = 10 printers.
- We need to calculate P(X ≥ 9), which is the sum of P(X=9) and P(X=10).
P(X=9) = (10C9) * 0.8^9 * (1-0.8)^(10-9)
P(X=9) = 10 * 0.8^9 * 0.2^1
P(X=10) = (10C10) * 0.8^10 * (1-0.8)^(10-10)
P(X=10) = 1 * 0.8^10 * 0.2^0
Finally, we can calculate P(X ≥ 9) by summing up these probabilities:
P(X ≥ 9) = P(X=9) + P(X=10)
For the second part of the question, if we consider 5 months with 10 units sold each month, we can treat each month independently and use the same calculations as in the first part for each month. To find the probability that at least 9 units operate correctly in each of the 5 months, we can calculate P(X ≥ 9) for each month and multiply them together.
I hope this explanation helps you understand how to solve these probability problems using the binomial distribution formula.