Question

An electronic message consists of a string of bits (0’s and 1’s). The message must pass through two relays before being received. At each relay there is a probability of .1 that the bit will be reversed before being relayed. Assume that the relays operate independently and the bits are independent.

(a) Find the probability that a single bit is received correctly.

(b) Find the probability that at least one bit of an eight-bit message is received in error.

Please explain step by step!

Answer 1

P(bit will be reversed before being relayed) = 0.1

a)

Required probability = P(a single bit is received correctly) = P(Not reveresed at 1st relay, Not reveresed at 2nd relay) + P(Reveresed at 1st relay, Reveresed at 2nd relay)

= (1-0.1)(1-0.1) + (0.1)(0.1)

= 0.82

b)

Let X denote the number of bits received in error. Then, $X\sim Bin(n=8,p=1-0.82=0.18)$

$P(X=x)=\binom{8}{x}(0.18)^{x}(1-0.18)^{8-x},x=0,1,2,.....,8$

Required probability = P(at least one bit of eight-bit message is received correctly) = $P(X\geq 1)=1-P(X=0)=1-\binom{8}{0}(0.18)^{0}(1-0.18)^{8-0}=0.7956$