# negative binomial

• Jun 26th 2008, 04:55 PM
amor_vincit_omnia
negative binomial
The probability that a machine produces a defective item is 0.01. Each item is check as it's produced. Assume these are independent trials and compute the probability that at least 100 items must be checked to find 1 defective item.

---
I let x= 100, r= 1,

so

99C0 * (0.01)(0.99)^99, which gives 0.003967

• Jun 26th 2008, 05:06 PM
bleesdan
I think what you found is the wrong probability.
What you are looking for is the probability that, after 99 items have gone by, you still haven't found a defective one. Then, anything that comes after it is irrelevant, because the first time you find a defective one, you will have checked at least 100 items.

So, you want to find the probability that you get 99 undefective items.
• Jun 26th 2008, 05:12 PM
amor_vincit_omnia
Well, if I dont multiply by the 0.01 I get the right answer.
But that's not what the question asks, or it doesn't seem so to me.
• Jun 26th 2008, 05:20 PM
bleesdan
Why are you multiplying by .01?
• Jun 26th 2008, 05:38 PM
amor_vincit_omnia
It's the formula for a negative binomial.
• Jun 26th 2008, 05:39 PM
amor_vincit_omnia
Formula for a negative binomial:

x-1 C r-1) (p^r)(1-p)^x-r
• Jun 26th 2008, 07:38 PM
CaptainBlack
Quote:

Originally Posted by amor_vincit_omnia
The probability that a machine produces a defective item is 0.01. Each item is check as it's produced. Assume these are independent trials and compute the probability that at least 100 items must be checked to find 1 defective item.

---
I let x= 100, r= 1,

so

99C0 * (0.01)(0.99)^99, which gives 0.003967

The probability that at least $100$ must be checked is the probability that the first $99$ are not defective, which is $0.99^{99} \approx 0.3697$

RonL
• Jun 26th 2008, 08:44 PM
mr fantastic
Quote:

Originally Posted by amor_vincit_omnia
The probability that a machine produces a defective item is 0.01. Each item is check as it's produced. Assume these are independent trials and compute the probability that at least 100 items must be checked to find 1 defective item.