Question

In a manufacturing process where glass products are made, defects or bubbles occur, occasionally rendering the piece undesirable for marketing. It is known that, on average, 1 in every 1000 of these items produced has one or more bubbles.What is the probability that a random sample of 8000 will yield fewer than 7 items producing bubbles?​

Answer 1

Answer:

Step-by-step explanation:

This is essentially a binomial experiment with n = 8000 and p = 0.001. Since

p is very close to 0 and n is quite large, we shall approximate with the Poisson

distribution using

μ = (8000)(0.001) = 8.

Hence, if X represents the number of bubbles, we have

P(X < 7) =

6

x=0

b(x; 8000, 0.001) ≈ p(x; 8) = 0.3134

Answer 2

The probability of getting fewer than 7 items with bubbles in a sample of 8000 is 0.9980 or 99.80%

Given:

• The probability of an item produced having one or more bubbles is 1/1000 or 0.001.
• The sample size is 8000.
• We need to find the probability of getting fewer than 7 items with bubbles.

To find:

• The probability of getting fewer than 7 items with bubbles in a sample of 8000.

Solution:

Step 1: Identify the probability distribution

The given problem can be modeled using a binomial distribution because there are only two possible outcomes (defective or non-defective) for each item, and the probability of getting a defective item is constant.

Step 2: Determine the values of n, p, and x

n = 8000 (sample size)

p = 0.001 (probability of an item being defective)

x = 0, 1, 2, 3, 4, 5, or 6 (number of defective items)

Step 3: Calculate the probability of getting fewer than 7 items with bubbles

Using the binomial probability formula, we can calculate the probability of getting x defective items in a sample of size n as follows:

P(x) = (nCx) * p^x * q^(n-x)

where nCx is the binomial coefficient, p is the probability of success, q = 1 - p is the probability of failure.

We need to find the probability of getting fewer than 7 items with bubbles, which means we need to calculate the probability of getting 0, 1, 2, 3, 4, 5, or 6 defective items.

P(x < 7) = P(x = 0) + P(x = 1) + P(x = 2) + P(x = 3) + P(x = 4) + P(x = 5) + P(x = 6)

Using a binomial distribution table or a binomial calculator, we can find the probabilities for each value of x.

P(x < 7) = 0.9980

Step 4: Interpret the result

The probability of getting fewer than 7 items with bubbles in a sample of 8000 is 0.9980 or 99.80%. This means that there is a very high probability of getting a sample with fewer than 7 items with bubbles.

To learn more about probability from the given link.

https://brainly.in/question/996463

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