In a binary communication system, a logic "1" occurs with probability p and is transmitted as the symbol S1 = A, while a logic "0" is transmitted as the symbol S0 = 0. If X is the random variable representing the transmitted sequence of bits, X can be represented as follows:
{S1 = A logic "1" P[X = S1] = p
S0 = -A logic "0" P[X = So] = 1 - p}
Noise is added to the signal during transmission so that the received base band signal can be expressed as Y = X + Z, where Z represents the noise signal which is assumed to be Gaussian (normal) with zero mean and standard deviation σ.

If the maximum a posteriori probability, MAP decision rule is used to decide which symbol has been transmitted
P[S1|Y] S1>|| If p = 0.5, find the probability of error.