A manufacturing company produces 10,000 plastic glasses per week. This company supplies plastic glasses to another company, which packages the glasses as part of picnic sets. The second company randomly samples 10 glasses from the supplier. If at least eight of the sampled glasses are non-defective, the second company accepts the lots. What is the probability that the lot will be not accepted if the glass manufacturing company is producing glasses that are 90% non-defective?

Question

A manufacturing company produces 10,000 plastic glasses per week. This company supplies plastic glasses to another company, which packages the glasses as part of picnic sets. The second company randomly samples 10 glasses from the supplier. If at least eight of the sampled glasses are non-defective, the second company accepts the lots. What is the probability that the lot will be not accepted if the glass manufacturing company is producing glasses that are 90% non-defective?

Answer 1

To solve this problem, we need to use the binomial probability formula.

The probability of having at least eight non-defective glasses out of ten can be calculated as follows:

P(X ≥ 8) = P(X = 8) + P(X = 9) + P(X = 10)

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

In this case:
n = 10 (total number of glasses sampled)
k = 8, 9, 10 (number of non-defective glasses)
p = 0.9 (probability of a single glass being non-defective)

Using the binomial formula, we can calculate the probabilities:

P(X = 8) = 10C8 * (0.9^8) * (0.1^2)
P(X = 9) = 10C9 * (0.9^9) * (0.1^1)
P(X = 10) = 10C10 * (0.9^10) * (0.1^0)

To calculate 10C8, 10C9, and 10C10, we can use the formula:

nCk = n! / (k! * (n-k)!)

Now let's calculate each probability:

P(X = 8) = 10! / (8! * (10-8)!) * (0.9^8) * (0.1^2)
= 45 * (0.9^8) * (0.1^2)

P(X = 9) = 10! / (9! * (10-9)!) * (0.9^9) * (0.1^1)
= 10 * (0.9^9) * (0.1^1)

P(X = 10) = 10! / (10! * (10-10)!) * (0.9^10) * (0.1^0)
= (0.9^10)

Now, we can calculate the probabilities:

P(X ≥ 8) = P(X = 8) + P(X = 9) + P(X = 10)
= 45 * (0.9^8) * (0.1^2) + 10 * (0.9^9) * (0.1^1) + (0.9^10)

Therefore, the probability that the lot will not be accepted is given by:

P(X < 8) = 1 - P(X ≥ 8)
= 1 - (45 * (0.9^8) * (0.1^2) + 10 * (0.9^9) * (0.1^1) + (0.9^10))

Simplifying the equation will give you the final probability.

Answer 2

To find the probability that the lot will be not accepted, we need to determine the probability of having less than eight non-defective glasses out of a random sample of ten.

First, we calculate the probability of selecting a non-defective glass. We are given that the glasses produced by the manufacturing company are 90% non-defective, so the probability of selecting a non-defective glass is 0.9.

Next, we can use the binomial probability formula to calculate the probability of having less than eight non-defective glasses in a sample of ten.

The binomial probability formula is:
P(X = k) = C(n, k) * p^k * (1 - p)^(n - k)

Where:
- P(X = k) is the probability of getting exactly k successes,
- C(n, k) is the combination formula (n choose k) and calculates the number of ways to choose k items from a set of n,
- p is the probability of success on a single trial,
- k is the number of successes in the sample, and
- n is the sample size.

In this case, we want to calculate P(X < 8), which is the probability of having less than eight non-defective glasses.

P(X < 8) = P(X = 0) + P(X = 1) + ... + P(X = 7)

P(X < 8) = Σ (P(X = k)) for k = 0 to 7

Now we can calculate each term of the summation using the binomial probability formula.

P(X < 8) = P(X = 0) + P(X = 1) + ... + P(X = 7)
= C(10, 0) * 0.9^0 * 0.1^10 + C(10, 1) * 0.9^1 * 0.1^9 + ... + C(10, 7) * 0.9^7 * 0.1^3

Now you can calculate each term using a combination calculator or manually calculating each term. After finding the probabilities for each term, sum them up to get the final probability that the lot will be not accepted, which is P(X < 8).