A channel with unequal inputs

Topic: Random experiments and probability · more questions on this topic
‹ Information about one die

A binary channel sends a 1 with probability 1/4 and a 0 with probability 3/4. A sent 1 is received as 1 with probability 0.90, and a sent 0 is received as 0 with probability 0.95.

  1. Find the probability that a 1 is received.

  2. Given that a 1 is received, find the probability that a 1 was sent.

Let S be the symbol sent and R the symbol received.

  1. By the law of total probability,

\mathbb{P}(R=1)=\mathbb{P}(R=1\mid S=1)\mathbb{P}(S=1)+\mathbb{P}(R=1\mid S=0)\mathbb{P}(S=0)=0.90\times\tfrac14+0.05\times\tfrac34=0.225+0.0375=0.2625.

  1. By Bayes’ rule,

\mathbb{P}(S=1\mid R=1)=\frac{0.225}{0.2625}=\frac67=0.857.

One received 1 in seven is a corrupted 0, because 0s are sent three times as often.