Bottomline Ink, a forms company, fills 100 orders a day with a 2% error rate in the completed orders. Assume this to be a Binomial Distribution. What is the probability that there will be more than 5 errors in a given day?
To find the probability that there will be more than 5 errors in a given day, we need to use the binomial distribution formula. In this case, the probability of success (p) is the error rate in completed orders, which is 2% or 0.02, and the number of trials (n) is the number of orders filled in a day, which is 100.
The formula for the probability mass function of a binomial distribution is:
P(X = k) = (n choose k) * (p^k) * ((1-p)^(n-k))
Where:
- P(X = k) is the probability of having exactly k successes (errors) in n trials (orders).
- (n choose k) is the binomial coefficient, which represents the number of ways to choose k successes from n trials.
- p^k represents the probability of k successes (errors).
- (1-p)^(n-k) represents the probability of (n-k) failures (correct orders).
To find the probability of having more than 5 errors, we need to calculate the sum of probabilities for k = 6, 7, 8, ..., 100. We can use the complement rule to calculate this probability.
P(X > 5) = 1 - P(X ≤ 5)
Let's calculate the probability step by step:
1. Calculate the probability of having 0 to 5 errors:
P(X ≤ 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
2. Use the binomial distribution formula to calculate each term:
P(X = k) = (100 choose k) * (0.02^k) * (0.98^(100-k))
3. Sum the probabilities for k = 0 to 5:
P(X ≤ 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
4. Finally, subtract the result from 1 to get the probability of having more than 5 errors:
P(X > 5) = 1 - P(X ≤ 5)
I will now calculate the probabilities step by step using the given information.