What is the expression for the error probability

Question

Solve the following problem:

In a binary communication system over an additive white Gaussian noise channel, two messages represented by antipodal signals s₁(t) and s₂(t) = -s₁(t) are transmitted. The probabilities of the two messages are p and 1 - p, respectively, where 0 ≤ p ≤ 1/2. The energy content of the each message is denoted by E, and the noise power spectral density is N₀/2.

1. What is the expression for the threshold value r_th such that f_orr > r_th the optimal detector makes a decision in favor of s₁(t)? What is the expression for the error probability?

2. Now assume that with probability of 1/2 the link between the transmitter and the receiver is out of service and with a probability of 1/2 this link remains in service. When the link is out of service, the receiver receives only noise. The receiver does not know whether the link is in service. What is the structure of the optimal receiver in this case? In particular, what is the value of the threshold rth in this case? What is the value of the threshold if p = 1/2? What is the resulting error probability for this case (p = 1/2)?