Binary digits, that is, 0’s and 1’s, are sent down a noisy communications channe

Question

Binary digits, that is, 0’s and 1’s, are sent down a noisy communications channel. They are received as sent with probability 0.9 but errors occur with probability 0.1. Assuming that 0’s and 1’s are equally likely to be sent. [Hint: let S be the sent digit and R be the received digit. Then we have that the sample space is partitioned by the two events {S = 0} and {S = 1}. We are also given that P(S = 0) = P(S = 1) = 0.5, P(R = 1|S = 1) = P(R = 0|S = 0) = 0.9, P(R = 0|S = 1) = P(R = 1|S = 0) = 0.1 ]

(a) What is the probability that a received digit is 1, P(R = 1)? [Hint: consider law of total probability P(R = 1) = P(S = 0)P(R = 1|S = 0) + P(S = 1)P(R = 1|S = 1). ]

NOTE: THIS IS NOT THE SAME QUESTION AS: "What is the probability that a 1 was sent given that we received a 1"

Explanation / Answer

HERE THE CHANCES OF SENDING THE DIGIT 0 OR DIGIT 1 HAS EQUAL CHANCES = 0.5

NOW WE NEED TO FIND THE PROBABILITY THAT DIGIT 1 IS RECEIVED.

WHICH CAN BE DONE IN TWO TYPES WHEN DIGIT 1 IS SEND OR DIGIT 0 IS SEND

WE HAVE TO ADD BOTH

NOW THE REQUIRED ANSWER WILL BE = P(S = 0)P(R = 1|S = 0) + P(S = 1)P(R = 1|S = 1)

= 0.5*0.1 + 0.5*0.9 = 0.5

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