Question

1. A transmitted message bit B is either 1 or 0, each with equal likelihood.

The bit is captured by N independent receivers; each receiver has a

probability p of receiving the wrong bit value (e.g. receiving a 1 when

a 0 was sent).

Write the conditional probability that the message bit is 1, if all N

receivers receive a 1. Write your answer as a function of p.

Answer 1

Here we are given that:

P(original 0) = P( original 1 ) = 0.5

Also, we are given that:

P( 1 received | original 0 ) = p and P( 1 received | original 1 ) = 1 - p

Therefore using law of total probability, for each receiver we have:

P( 1 received ) = P( 1 received | original 0 )P(original 0) + P( 1 received | original 1 ) P( original 1 )

P( 1 received ) = 0.5p + 0.5(1 - p) = 0.5

Now for N such receivers

P( 1 received by all | original 1 ) = (1-p)^N and P( 1 received by all | original 0 ) = p^N

Using law of total probability again, we get here:

P( 1 received by all ) = 0.5( 1- p )^N + 0.5 p^N

Therefore now using bayes theorem, we get here:

This is the required expression for the required probability.