Question

Suppose that three active nodes a, b, and c are competing to send an infinite number of packets. Each node attempts to transmit in each slot with probability p. The first slot is numbered slot 1, the second slot is numbered slot 2, and so on. What is the probability that node a succeeds for the first time in slot 5?

Answer 1

The probability that node a succeeds for the first time in slot 5 is q^4 * p.

Step-by-step explanation:

To find the probability that node a succeeds for the first time in slot 5, we need to consider the probability that nodes b and c fail in the first four slots, and node a succeeds in the fifth slot.

The probability of node a succeeding in a slot is p, and the probability of node a failing in a slot is q = 1-p.

Therefore, the probability that nodes b and c both fail in the first four slots is q*q*q*q = q^4.

The probability that node a succeeds in the fifth slot is p.

So, the probability that node a succeeds for the first time in slot 5 is q^4 * p.