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There are n stations in a 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?
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- np(1 – p)n – 1
- (1 – p)n – 1
- p(1 – p)n – 1
- 1 – (1 – p)n – 1
- np(1 – p)n – 1
Correct Option: A
It is already given that n stations transmit with a probability p in a slotted LAN. For this, the stations need to acquire a medium. Let us assume the following
1 be the event when station 1 transmits
2 be the event when station 2 transmits
N be the event when station N transmits The probability for a station to transmit is represented as
P( 1 2 3 ......N ∪ 1 2 3 4 ......N ∪ 1 2 3 4 5 ......N ∪ ......)
As all the events are mutually exclusive. Hence, the probability can be given as
P(1) P(2) P(3) ...... P(N) + P(1) P(2) P(3) P(4) ...... P(N) + P(1) P(2) P(3) P(4) P(5) ...... P(N) + ........
As given that all events are independent, hence the probability
= p(1 – p) (1 – p)... (1 – p) + (1 – p) p (1 – p)... (1 – p) + (1 – p) (1 – p) (1 – p)... (1 – p) + ....
= np (1 – p) n – 1
