Probability in a Communication Channel

P(A)

0.6

P(X|A)

0.9

P(Y|A)

0.9

P(A) = 0.6	P(B) = 0.4
P(X)P(B)P(Y\|A)+P(A)P(X\|A) = 0.58	P(Y)P(A)P(X\|B)+P(B)P(Y\|B) = 0.42
P(A\|X) P(A)P(X\|A) P(X) = 0.93	P(B\|Y) P(B)P(Y\|B) P(Y) = 0.86
P(B\|X) P(B)P(X\|B) P(X) = 0.07	P(A\|Y) P(A)P(Y\|A) P(Y) = 0.14

In a binary communication system, the symbols 0 and 1 are sent through a channel, but noise can switch 1 to 0 or vice versa with certain probabilities. The error can be estimated by Bayes' rule.

Let

A

be the event of sending 0 (represented by

A0

) and

B

be the event of sending 1 (represented by

B0

), with corresponding probabilities

P(A)

and

P(B)

. On the receiving side,

X

and

Y

are the events of receiving 0 or 1. Represented by

X←0

and

Y←1

,

P(X|A)

is the probability of receiving 1 given that 0 was sent,

P(Y|B)

is the probability of receiving 1 given that 1 was sent,

P(A|X)

is the conditional probability of

A

given

X

, and so on.