A sequence of relay stations 1,...,6 are arranged in a line. A relay station can

Question

A sequence of relay stations 1,...,6 are arranged in a line. A relay station can communicate with any other station at most 2 away. So station 4 can only communicate with stations 2, 3, 5, and 6, for example. The relays, however, have not been tested, and it is known that each relay is working only with a 90% chance. Standing at station 1, what is the probability that someone at station 6 will recieve your message? Note: both stations 1 and 6 need to be working for the message to go through. (Challenge, not for credit: what if there are n stations? Hint: find a recurrence relation.)

Explanation / Answer

In order for the message to be delivered to the final station number 6, the message can go either by passing through each station, or by making a jump which means skipping the adjacent station.

Let X denote the number of skips made by the message until it reaches station number 6, starting from 1.

We have:

P(X=0) = 0.96 = 0.531

P(X=1) = 4C1*(0.95)*0.1 = 0.236

P(X=2) = 2*(0.94)*(0.1)2 = 0.013

So,

Reqd probability = 0.531+0.236+0.013 = 0.78

Hope this helps !

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