1

A sample of size $r$ is taken from a population of $n$ elements. Find the probability that none
of the $N$ prescribed elements will be included in the sample, assuming the sampling to be

a) without
b) with replacement

.


Compare the numerical value for the two methods, when

(i) n=100, r=N=3
and
(ii) n=100, r=N=10

.

Brian makenzy
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  • Brian - it is usually considered "good manners" on the site that when you receive a helpful answer, you accept it. You can accept one answer per question asked. To accept an answer, you simply click on the $\large\checkmark$ to the left of the answer you'd like to accept. It turns green when you click on it. Plus, you get +2 reputation points each time you accept an answer. But remember, you can only accept one answer per question. – amWhy Sep 26 '13 at 14:09

1 Answers1

1

Without replacement: We can choose $r$ items in $\binom{n}{r}$ equally likely ways.

There are $\binom{n-N}{r}$ to do it using only allowed numbers. Divide.

With replacement: The probability that on any pick we miss all the $N$ "bad" ones is $\frac{n-N}{n}$. For the probability of avoiding bad ones $r$ times in a row, take the $r$-th power.

André Nicolas
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