Intereting Posts

Show that series converge or diverge
Derivative (or differential) of symmetric square root of a matrix
What is a function to represent a diagonal sine wave?
Complex analysis book for Algebraic Geometers
Using contour integration, or other means, is there a way to find a general form for $\int_{0}^{\infty}\frac{\sin^{n}(x)}{x^{n}} \, dx$?
A determinant inequality
A definite integral related to Ahmed's integral
Indefinite Integral with “sin” and “cos”: $\int\frac{3\sin(x) + 2\cos(x)}{2\sin(x) + 3\cos(x)} \; dx $
Is it possible to evaluate $ \int_0^1 x^n \, dx$ by contour integration?
Why can't I use the disk method to compute surface area?
Find shortest primary decomposition.
Can a cube always be fitted into the projection of a cube?
Prove that $ n < 2^{n}$ for all natural numbers $n$.
Which trigonometric identities involve trigonometric functions?
Are there any sets other than the usual in which we can apply Sturm's axioms?

In sampling without replacement the probability of any fixed element in the population to be included in a random sample of size $r$ is $\frac{r}{n}$. In sampling with replacement the corresponding probability is $\left[1- \left(\frac{1}{1-n}\right)^r\right]$.

Please help me show how this is proved.

- Number of binary strings of length 8 that contain either three consecutive 0s or four consecutive 1s
- Relation between exclusive-OR and modular addition in a specific function
- Number of bit strings with 3 consecutive zeros or 4 consecutive 1s
- Combinatorics Pigeonhole problem
- Is there an elegant bijective proof of $\binom{15}{5}=\binom{14}{6}$?
- Number of binary and constant-weight strings of length m which do not contain k consecutive zeros

- union of two independent probabilistic event
- There exists a vector $c\in C$ with $c\cdot b=1$
- Probability vs Confidence
- Enigma : of Wizards, Dwarves and Hats
- Given n ranging from 1 to 100, find sum of digits equal to half of arithmetic sum of 1 to 100
- Combinations of selecting $n$ objects with $k$ different types
- Monochromatic Rectangle of a 2-Colored 8 by 8 Lattice Grid
- Cardinality of the set $D$
- Binomial probability with summation
- A formula for heads and tail sequences

Just a note in terms of nomenclature:

$$

{n \choose r} = {_n}C_r = \frac{n!}{r!(n-r)!}

$$

There are ${n \choose r}$ ways to select the sample of $r$ elements from the pool of $n$ items. That is our denominatorâ€”the universe of possible results. To count the number of samples which *include* our special fixed element (call it $x^*$) we can just realize that we **must** pick $x^*$, and there is only one way to do that (${1 \choose 1}$, and that leaves us a pool of $n-1$ item from which we need to select $r-1$ to fill out the sample. There are:

$$

{1 \choose 1}{n-1 \choose r-1}

$$

ways to create our desired samples. So the probability of having $x^*$ in our mix is:

$$

\frac{{n-1 \choose r-1}}{{n \choose r}} = \frac{(n-1)!}{(r-1)!(n-r)!}\div\frac{n!}{r!(n-r)!}= \frac{(n-1)!}{(r-1)!(n-r)!}\cdot\frac{r!(n-r)!}{n!} = \mathbf{\frac{r}{n}}

$$

First a clarification. When sampling without replacement, the maximum number of times $x^*$ can appear is, of course, $1$. When sampling **with** replacement, it can appear between $0$ and $r$ times. Judging by the answer you gave, the question you want to answer is the number of ways the fixed element $x^*$ appears **at least** once. That is most easily addressed by realizing it is *all* possible ways *except* for the times that it *does not appear at all*. In other words:

$$

P(\textrm{at least once}) = 1 – P(\textrm{never})

$$

In order for $x^*$ to never appear, it cannot appear in any of the $r$ slots, which means that we can only pick from the remaining $n-1$ items. The probability that in slot $1$ we select something *other* than $x^*$ is $\frac{n-1}{n} = 1 – \frac{1}{n}$. This has to happen in **every one** of the $r$ slots, so the probability of having *no* manifestations of the fixed element in a sample of size $r$ is:

$$

\left(\frac{n-1}{n}\right)^r = \left(1 – \frac{1}{n}\right)^r

$$

So the probability of *at least* one showing is everything else, or:

$$

\mathbf{1- \left(1 – \frac{1}{n}\right)^r}

$$

Unfortunately, this is not the same as the value you posted in the question. Is it at all possible that the value in the innermost parentheses was supposed to be $1-\frac{1}{n}$ and not $\frac{1}{1-n}$?

**for without replacement :-**

total no. of possible ways of selecting $r$ elements from $n$ elements = $_nC_r$

total no. of ways where element $x$ is always selected would be equal to selecting $(r-1)$ element from $(n-1)$ elements [as we would consider $x$ to be already selected] ,

which would be = $_{(n-1)}C_{(r-1)}$

probability =

$$

\frac{{_{(n-1)}}C_{(r-1)}}{_nC_r} = \frac{r}{n}

$$

:-> $C$ is the combination

**for with replacement :-**

total possible no. of selections would be = C(n+r-1,r) [bars and star logic]

total cases where element x is never selected are = C(n+r-2,r) [n reduces to n-1]

probability of at least one selection of element x = [1 – {C(n+r-2,r)}/{C(n+r-1,r)}]

which comes out to be = (n-1)/(n+r-1)

- What are Free Objects?
- Integral $ \int_{-\infty}^\infty \frac{e^{ikx}}{x^{3/2}}dx$
- Finding the sum of the series $\frac{1}{1!}+\frac{1+2}{2!}+\frac{1+2+3}{3!}+ \ldots$
- Simple L'Hopital Question
- How to do contour integral on a REAL function?
- Intuition behind Taylor/Maclaurin Series
- Equivalence of geometric and algebraic definitions of conic sections
- Does the series $\sum_{n = 1}^{\infty}\left(2^{1/n} – 1\right)\,$ converge?
- Evaluate the double sum $\sum_{m=1}^{\infty}\sum_{n=1}^{m-1}\frac{ 1}{m n\left(m^2-n^2\right)^2}$
- Where can I learn about complex differential forms?
- An example for a calculation where imaginary numbers are used but don't occur in the question or the solution.
- Converging series question, Prove that if $\sum_{n=1}^{\infty} a_n^{2}$ converges, then does $\sum_{n=1}^{\infty} \frac {a_n}{n}$
- balance scale problem for 13 (not 12) items
- A Banach space of (Hamel) dimension $\kappa$ exists if and only if $\kappa^{\aleph_0}=\kappa$
- Is the integral of a measurable function measurable?