# Math

Suppose you work for the quality control department at Hewlett Packard. You know that ink jet cartridges are sometimes defective. The current defective rate is 8 cartridges per package of 100 (i.e. 0.08) What is the probability of sampling 12 cartridges and finding:

Less than 3 are defective:
More than 2 are defective:
At least 4 are defective:
Exactly 1 is defective:
2 or more, but less than 5 is defective:

If you could help me on at least 1 I would greatly appreciate that, or even point me where I need to go, I have no clue what to do here.

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1. this is a binomial probability ... defective or not defective

probability (d) = .08 , probability (n) = .92 ... (n + d)^12

expand the binomial and substitute the probabilities in the desired terms
... n^12 + 12 n^11 d + 66 n^10 d^2 + ... + 66 n^2 d^10 + 12 n d^11 + d^12

less than 3 ... .92^12 + (12 * .92^11 * .08) + (66 * .92^10 * .08^2)

p(more than 2) = 1 - p(less than 3)

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posted by scott
2. I'm so sorry, but is there any way you can break down the whole (12 * .92^11 * .08) + (66 * .92^10 * .08^2) part for me. I'm just not understanding where all of these values came from. aside from 0.92 and 0.08

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posted by Jaz
3. when you expand the binomial , the 12, 66, etc. are the coefficients of the terms

look up Pascal's Triangle for background

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posted by scott
4. So if I'm understanding this right, for the probability of it being less than 3 my answer should be 0.93 right?

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posted by Jaz
5. yes ... .9348

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posted by scott
6. More than 2 would be 0.06?

Can you walk me through what I need to do for the last 3 please?

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posted by Jaz
7. at least 4 ... 1 - (first 4 terms)

Exactly 1 ... second term

2 or more, but less than 5 ... (first 5 terms) - (first 2 terms)

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posted by scott
8. Less than 3 are defective
---> none defective + one defective + 2 defective
= C(12,0)(.08)^0 (.92)^12 + C(12,1)(.08)^1 (.92)^11 + C(12,2)(.08)^2 (.92)^10
= .367666... + .38365... + .18348..
= appr .9348

More than 2 are defective
---> 3 defective + 4 def + ... + 12 def.
or 1 - (none def + 1 def + 2 def)

= 1 - (C(12,0)(.92^0)(.08^12) + C(12,1)(.92^1)(.08^11) + C(12,2)(.92^2)(.08^10) )
= ...

do you get the idea?

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posted by Reiny

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