Probability Finding Binomial Average Problem

In summary, the conversation discusses a manufacturing process where a random sample of nine articles is taken from each batch, and if one article is defective, the batch is rejected. It is also mentioned that a second sample may be taken if the first sample results in a rejection. The question asks for the average number of articles sampled per batch when the proportion of defective articles, p, is 0.1. The conversation also includes a discussion on the process of finding the average number of errors using binomial probabilities.
  • #1
caljuice
70
0

Homework Statement



In each batch of manufactured articles, the proportion of defective articles is p. From each
batch, a random sample of nine is taken and each of the nine articles is examined. If one article is found to be defective, the batch is rejected; otherwise, it is accepted. If rejected, a second sample of nine is taken and also rejected if an error is found. Otherwise it is accepted.

Evaluate the average number sampled per manufactured batch over a large number of batches when p has the value 0.1. [4 marks] (Note: Just to be sure you understand what this is asking, you are asked to calculate the average number of articles sampled per manufactured batch over the long run.)

I condensed the question here is the full incase I worded it poorly:

In each batch of manufactured articles, the proportion of defective articles is p. From each
batch, a random sample of nine is taken and each of the nine articles is examined. If two
or more of the nine articles are found to be defective, the batch is rejected; otherwise, it
is accepted. Show that the probability that a batch is accepted is (1−p)
8
(1+8p). [3 marks]
It is decided to modify the sampling scheme so that when one defective is found in the
sample, a second sample of nine is taken (each of the nine articles is examined) and the
batch rejected if this contains any defectives. With this exception, the original scheme is
continued. Find an expression in terms of p for the probability that a batch is accepted.
[3 marks].
For this modified scheme, evaluate the average number sampled per manufactured batch
over a large number of batches when p has the value 0.1. [4 marks] (Note: Just to be
sure you understand what this is asking, you are asked to calculate the average number of
articles sampled per manufactured batch over the long run.)
Attempt:
Any hints on the process?

My guess:

The probability for the # of error is easy using binomials.

It's discrete binomial question so I was thinking about making a table to find E(X) where X = # of error,

average # error = E(X) = X1*P1 + X2*P2 ... X18*P18 , 18 since 2 samples and professor hint was answer is between 9 - 18.

E(X) = np since binomial, where n = 18?
E(X) / p = n , which is the answer?

Thanks for any help.
 
Last edited:
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  • #2
caljuice said:

Homework Statement



In each batch of manufactured articles, the proportion of defective articles is p. From each
batch, a random sample of nine is taken and each of the nine articles is examined. If one article is found to be defective
Do you mean "at least one"?

, the batch is rejected; otherwise, it is accepted. If rejected, a second sample of nine is taken and also rejected if an error is found. Otherwise it is accepted.
What happens if the batch is rejected a second time? Do you do only one or two samples?

Evaluate the average number sampled per manufactured batch over a large number of batches when p has the value 0.1. [4 marks] (Note: Just to be sure you understand what this is asking, you are asked to calculate the average number of articles sampled per manufactured batch over the long run.)

I condensed the question here is the full incase I worded it poorly:

In each batch of manufactured articles, the proportion of defective articles is p. From each
batch, a random sample of nine is taken and each of the nine articles is examined. If two
or more of the nine articles are found to be defective, the batch is rejected; otherwise, it
is accepted. Show that the probability that a batch is accepted is (1−p)
8
(1+8p). [3 marks]
It is decided to modify the sampling scheme so that when one defective is found in the
sample, a second sample of nine is taken (each of the nine articles is examined) and the
batch rejected if this contains any defectives. With this exception, the original scheme is
continued. Find an expression in terms of p for the probability that a batch is accepted.
[3 marks].
For this modified scheme, evaluate the average number sampled per manufactured batch
over a large number of batches when p has the value 0.1. [4 marks] (Note: Just to be
sure you understand what this is asking, you are asked to calculate the average number of
articles sampled per manufactured batch over the long run.)
Attempt:
Any hints on the process?

My guess:

The probability for the # of error is easy using binomials.

It's discrete binomial question so I was thinking about making a table to find E(X) where X = # of error,

average # error = E(X) = X1*P1 + X2*P2 ... X18*P18 , 18 since 2 samples and professor hint was answer is between 9 - 18.

E(X) = np since binomial, where n = 18?
E(X) / p = n , which is the answer?

Thanks for any help.
 
  • #3
Sorry I meant at least one. I was a little confused on the initial wording too, but I believe it is a max of two samples.
 
  • #4
I see someone else is also doing the STAT 251 HW late. :-p
 

FAQ: Probability Finding Binomial Average Problem

1. What is the formula for finding the binomial average in a probability problem?

The formula for finding the binomial average, also known as the mean or expected value, is n * p, where n is the number of trials and p is the probability of success in each trial.

2. How do you determine the number of trials in a binomial probability problem?

The number of trials in a binomial probability problem is determined by the number of independent events or trials that occur. For example, if you are flipping a coin 10 times, the number of trials would be 10.

3. What is the difference between binomial probability and binomial distribution?

Binomial probability refers to the likelihood or chance of a specific outcome occurring in a series of independent trials. Binomial distribution, on the other hand, refers to the pattern or distribution of these outcomes over a large number of trials.

4. Can the binomial average be a decimal number?

Yes, the binomial average can be a decimal number. This is because it is a calculated value based on the number of trials and the probability of success, which can both be decimal numbers.

5. How can the binomial average be used in real-life situations?

The binomial average can be used in real-life situations to determine the expected number of successes in a series of trials. For example, it can be used in business to predict the success rate of a marketing campaign, or in genetics to predict the likelihood of inheriting a certain trait.

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