Q:

Probability of error of a system?

Let bits 0 and 1 be equi probable. For bit 0, any value between -0.3 V and 0.3 V can be taken
with equal probability. Similarly for bit 1, any value between 0 V and 1 V can be taken with
equal probability. If threshold is 0.25 V i.e. bit 1 is detected if received voltage is greater
than 0.25 V and bit 0 is detected otherwise, then find the probability of error of this system.

A:

Using the Bayes Theorem, where $B$ is the bit sent and $V$ is the received voltage):
$$P(E) = P(B = 0)P(V > 0.25| B = 0) + P(B = 1)P(V<0.25|B = 1)$$
$$P(E) = 0.5\times((0.3-0.25)/(0.3-(-0.3)) + 0.5\times(0.25-0) = 1/6$$

