A one received on a noisy channel

Topic: Random experiments and probability · more questions on this topic
‹ Five balls out of twenty A zero received ›

A binary channel sends a 1 with probability 2/3 and a 0 with probability 1/3. A sent 1 is received as 1 with probability 0.95. A sent 0 is received as 0 with probability 0.90. A 1 is received. Find the probability that a 1 was sent.

Worked answer (click to show)

Let B be the event that a 1 was sent and A the event that a 1 was received. Then \mathbb{P}(A\mid B)=0.95, \mathbb{P}(A\mid B^c)=0.10, \mathbb{P}(B)=2/3 and \mathbb{P}(B^c)=1/3. Bayes’ rule for events gives

\mathbb{P}(B\mid A)=\frac{0.95\times 2/3}{0.95\times 2/3+0.10\times 1/3}=\frac{1.9}{2}=0.95.