Question

In a binary communication channel, 0s and 1s are transmitted with equal probability. The probability that a 0 is correctly received (as a 0) is 0.99. The probability that a 1 is correctly received (as a 1) is 0.90. Suppose we receive a 1, what is the probability that, in fact, a 0 was sent?

How to apply bayes rule?

Answer 1

Since digits are transmitted with equal probability so

P(sent 0) = P(sent 1) = 0.5

From given information we have

P(received 0 | sent 0) = 0.99

P(received 1 | sent 1) = 0.90

By the complement rule we have

P(received 1 | sent 0) = 1-P(received 0 | sent 0) = 0.01

We need to find the probability P(sent 0 | receive 1). By the Baye's theorem we have

P(sent 0 | receive 1) = [P(receive 1 |sent 0) P(sent 0) ] / [ P(receive 1 |sent 0) P(sent 0)+ P(receive 1 |sent 1) P(sent 1) ] = [ 0.01 * 0.50] / [ 0.01*0.50 + 0.90 * 0.50] = 0.01 / 0.91 = 0.010989