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?
The probability that the lot will not be accepted is 0.001.
To find the probability that the lot will not be accepted, we first need to determine the probability of having less than eight non-defective glasses in a sample of 10.
The manufacturing company produces glasses that are 90% non-defective, so the probability of selecting a non-defective glass is 0.9.
We can use the binomial distribution formula to calculate the probability. The formula is:
P(X = k) = (n choose k) * p^k * q^(n-k)
Where:
P(X = k) is the probability of getting exactly k successes,
n is the number of trials (sample size),
k is the number of successful trials (non-defective glasses),
p is the probability of success on a single trial (non-defective glasses),
q is the probability of failure on a single trial (defective glasses), which is calculated as 1 - p.
In this case, n = 10, p = 0.9, and q = 1 - p = 0.1.
Now, we need to calculate the probability of having less than eight non-defective glasses, which means k = 0, 1, 2, 3, 4, 5, 6, and 7.
P(X < 8) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7)
We can calculate each term using the binomial distribution formula and sum them up to find the final probability.
P(X = 0) = (10 choose 0) * 0.9^0 * 0.1^10
P(X = 1) = (10 choose 1) * 0.9^1 * 0.1^9
P(X = 2) = (10 choose 2) * 0.9^2 * 0.1^8
P(X = 3) = (10 choose 3) * 0.9^3 * 0.1^7
P(X = 4) = (10 choose 4) * 0.9^4 * 0.1^6
P(X = 5) = (10 choose 5) * 0.9^5 * 0.1^5
P(X = 6) = (10 choose 6) * 0.9^6 * 0.1^4
P(X = 7) = (10 choose 7) * 0.9^7 * 0.1^3
Finally, sum up all the probabilities to find P(X < 8).
P(X < 8) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7)
To calculate the probability that the lot will not be accepted, we need to determine the probability that less than 8 out of the 10 sampled glasses are non-defective.
Let's break down the problem step by step:
Step 1: Determine the probability of a non-defective glass.
Given that the glasses manufactured by the company are 90% non-defective, the probability of a single glass being non-defective is 0.90 (90%).
Step 2: Calculate the probability of at least 8 non-defective glasses.
To find this probability, we need to consider all possible combinations of 8, 9, or 10 non-defective glasses out of the 10 sampled glasses.
- Number of ways to choose 8 non-defective glasses out of 10: C(10, 8) = 45
The formula for combination (nCr) is n! / (r!(n-r)!).
In this case, it's 10! / (8!(10-8)!) = 45.
- Number of ways to choose 9 non-defective glasses out of 10: C(10, 9) = 10
In this case, it's 10! / (9!(10-9)!) = 10.
- Number of ways to choose 10 non-defective glasses out of 10: C(10, 10) = 1
In this case, it's 10! / (10!(10-10)!) = 1.
Now, we need to calculate the probability of each combination occurring, assuming independence:
Probability of 8 non-defective glasses: (0.90)^8
Probability of 9 non-defective glasses: (0.90)^9
Probability of 10 non-defective glasses: (0.90)^10
Step 3: Add up the probabilities.
Now, we need to add up the probabilities of each combination occurring:
Probability of accepting the lot = Probability of at least 8 non-defective glasses
= (Probability of 8 non-defective glasses) + (Probability of 9 non-defective glasses) + (Probability of 10 non-defective glasses)
= (0.90)^8 + (0.90)^9 + (0.90)^10
Step 4: Calculate the probability of not accepting the lot.
Finally, to get the probability of not accepting the lot, subtract this probability from 1:
Probability of not accepting the lot = 1 - Probability of accepting the lot
= 1 - [(0.90)^8 + (0.90)^9 + (0.90)^10]
Hence, to find the probability that the lot will not be accepted, you would calculate 1 - [(0.90)^8 + (0.90)^9 + (0.90)^10].