Combinatorics Problem: Finding Samples with Non-Conforming Chips

[Ownaginatious]

So here is the problem:

A lot of 140 semiconductor chips is inspected by choosing a sample of five chips. Assume 10 of the chips do not conform to customer requirements.

...

c) How many samples of five contain at least one non-conforming chip?

Now what seems logical to me is first choose 1 of the 10 non-performing and then choose 4 from the remaining 139 chips.

What is wrong with my logic here? I don't get the answer the book gets (130,721,752), and instead get 148, 916, 260.

 

[mXSCNT]

Suppose we simplify it a bit so we have 2 non-performing, A and B, and 2 performing, C and D, and we are looking at a sample size of 2. Using your method we first pick A from the 2 non-performing, and then 1 of the remaining 3, giving us:
AB
AC
AD
Now we pick B from the two non-performing, and 1 of the the remaining 3, giving us:
BA
BC
BD

Unfortunately we have counted AB twice, first as AB and then as BA. So that's why your way doesn't work.

To solve it correctly you should take the total number of samples, which is C(140,5), and then subtract the ones that don't have any non-performing elements, which is C(130,5).

 

[statdad]

"What is wrong with my logic here? I don't get the answer the book gets (130,721,752), and instead get 148, 916, 260."

The phrase "at least one" does not mean the same thing as "exactly one". The problem involved at least one, you answered as if it were exactly one.

 

[mXSCNT]

"What is wrong with my logic here? I don't get the answer the book gets (130,721,752), and instead get 148, 916, 260."

The phrase "at least one" does not mean the same thing as "exactly one". The problem involved at least one, you answered as if it were exactly one.

If you read closer, he actually made a different mistake. He said "choose 4 from the remaining 139 chips" not "choose 4 from the remaining 130 chips" (which would be the "exactly one" option).

 

[blue_raver22]

I would approach this problem using combinatorics, which is the branch of mathematics that deals with counting and arranging objects. In this case, we are interested in finding the number of samples of five chips that contain at least one non-conforming chip.

To solve this problem, we can use the concept of combinations. In combinatorics, a combination is a way of selecting a subset of objects without regard to their order. In this problem, we want to select 5 chips from a group of 140 chips, where order does not matter.

Using the formula for combinations, we can calculate the number of samples that contain at least one non-conforming chip as:

C(140,1) * C(10,1) * C(139,4) = 140 * 10 * 139 * 138 * 137 / (1 * 1 * 4 * 3 * 2) = 130,721,752

This is the same answer that the book gets. Your approach of first choosing 1 non-conforming chip and then choosing 4 from the remaining chips is incorrect because it does not take into account the fact that the order of the chips does not matter. In your calculation, you are counting the same sample multiple times, once for each possible order of the chips. This is why your answer is much larger than the correct answer.

In conclusion, to solve this problem, we need to use the concept of combinations and take into account that the order of the chips does not matter. By doing so, we arrive at the correct answer of 130,721,752 samples containing at least one non-conforming chip.