GATE | GATE-CS-2007 | Question 65

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?
(A) (1-p)(n-1)
(B) np(1-p)(n-1)
(C) p(1-p)(n-1)
(D) 1-(1-p)(n-1)


Answer: (B)

Explanation:  

P(X) = Probability that station X attempts to transmit = P
P (-X) = Probability that station X does not transmit = 1-P
Required is: Probability that only one station transmits = y

Y = (A1, -A2, -A3…… -An) + (-A1, A2, A3……-An) + (-A1, -A2, A3…..-An) + ……….. + (-A1, -A2, -A3……An)
= (p*(1-p)*(1-p)*…… (1-p) + (1-p)*p*(1-p)……..(1-p) + ………….
= p*(1-p)^(n-1) + p*(1-p)^n-1 + ……………………………………………. + p*(1-p)^(n-1)
= n*p*(1-p)^(n-1)

This solution is contributed Anil Saikrishna Devarasetty .

See question 3 of https://www.geeksforgeeks.org/computer-networks-set-9/

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