# You are the manager of The Copy Shop. For your big copy machine, the probability a copy of a...

## Question:

You are the manager of The Copy Shop. For your big copy machine, the probability a copy of a document will have a defect is {eq}0.1 {/eq}. ("A defect" means one or more defects.) What is the probability that the 7th copy will be the first one with a defect?

{eq}X {/eq} = number of the copy that is the first one with a defect

a. {eq}X \sim {/eq} binomial.

b. {eq}X \sim {/eq} negative binomial.

c. {eq}X \sim {/eq} hypergeometric.

d. {eq}X \sim {/eq} Poisson.

## Negative binomial distribution

Negative binomial distribution is the distribution which is used only to find the probability of the first success at the nth number. If before the nth number, first success is already got, then the geometric distribution is used instead of Negative binomial distribution

Given Information

Number of copies = 7

Probability of a defect in a copy = 0.1

Since, before the 7th number, there is no success, so the best distribution is Negative binomial distribution.

Let X is the number at which first defect occur.

The probability that the 7th copy will be the first one with a defect is given as:

{eq}\begin{align*} P\left( {X = 7} \right) &= {\left( {1 - p} \right)^{x - 1}}p\\ &= {\left( {1 - 0.1} \right)^6}\left( {0.1} \right)\\ &= 0.0531 \end{align*} {/eq}

Therefore, the probability that the 7th copy will be the first one with a defect is 0.0531

Hence, the correct option is b