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Suppose there are N stations in a slotted LAN. Each station attempts to transmit with a probability P in each time slot. The probability that only one station transmits in a given slot is____

1. $nP(1-P)^{n-1}$
2. $nP$
3. $P(1-P)^{n-1}$
4. $n^P (1-P)^{n-1}$

It can be derived using Binomial theorem  of probability

NCi((p)i(1-p)n-i

Now 1 stations is sending with probability P ,and n-1 stations are not sending with probability 1-p

Now the probaility is= nC1 (p)1(1-p)n-1

=np(1-p)n-1

so a is correct answer here

A) Np(1 – p)N-1.

the probability a  node has a success is p(1 – p)N-1. Because there are N nodes, the probability that any one of the N nodes has a success is Np(1 – p)N-1.

It can be derived using Binomial theorem  of probability

NCi((p)i(1-p)n-i

Now 1 stations is sending with probability P ,and n-1 stations are not sending with probability 1-p

Now the probaility is= nC1 (p)1(1-p)n-1

=np(1-p)n-1

so a is correct answer here

The probability that a particular station transmits and no body else transmits = $p*(1-p)^{(n-1)}$
The probability that any station can transmit = n*(probability that a particular station transmits)

= $\binom{n}{1}*p*(1-p)^{(n-1)}$

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