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There are $n$ stations in slotted LAN. Each station attempts to transmit with a probability $p$ in each time slot. What is the probability that ONLY one station transmits in a given time slot?

1. $np(1-p)^{n-1}$

2. $(1-p)^{n-1}$

3. $p(1-p)^{n-1}$

4. $1-(1-p)^{n-1}$

$\text{(A) is correct option!}$ From which book ??
Application of this question: https://gateoverflow.in/3729/gate2004-it-85
Book : Computer Networking - Kurose : Ross 7th Global edition

Page : 485

Probability that only one station transmits in a given slot = ${n \choose 1} p^1 (1-p)^{n-1}$

Answer is option A.

$p$ for 1 transmitting and $(1-p)$ for $n-1$ non transmitting and $n$ ways to choose $1$ from $n$.

Can you explain it more ?
Binomial distribution with parameter (n,p)
Please before making above question clear your concept about Binomial distribution,then this question will seem like birthday cake

Solotion::

P(X=1)=nC1 x (p)^1 x (1-p)^n-1

Let the stations be $S_1, S_2, S_3,...,S_n$

The probability that only one station transmits is:-

Either $S_1$ transmits and no other transmits

Or, $S_2$ transmits and no other transmits

Or, $S_3$ transmits and no other transmits

so on.

$p(1-p)^{n-1}+p(1-p)^{n-1}+p(1-p)^{n-1}+...$

=> $np(1-p)^{n-1}$

Option A

### 1 comment

Thanks!
There are "n" stations in LAN. According to question, only one station want to transmit , in the sense among "n" stations, "n-1" should not be allowed to transmit the data  when one station is already is tranmitting the data.

In our question suppose "n" leveled station is transmitting the data with probability "p" then remaining '"n-1" should not be allowed to transmit the data at given slot.

So, np.(1-p)^ n-1.

(B)

### 1 comment

(Building on what @akash.dinkar12 pointed)

it is a simple probability+PnC question, of type:

given n events E, P(E) = p and P(E') = 1-p,

Find the P(Success), where Success = exactly 1 event occurs.