Question

A binary communication channel transmits a sequence of "bits" (0s and 1s). Suppose that for any particular bit transmitted, there is a 15% chance of a transmission error (a 0 becoming a 1 or a 1 becoming a 0). Assume that bit errors occur independently of one another. (Round your answers to four decimal places.)

(a)

Consider transmitting 1000 bits. What is the approximate probability that at most 165 transmission errors occur?

(b)

Suppose the same 1000-bit message is sent two different times independently of one another. What is the approximate probability that the number of errors in the first transmission is within 60 of the number of errors in the second?

Answer 1

a)

here for expected number of errors =np=1000*0.15=150

and std deviation =sqrT(np(1-p))=11.29

hence from normal approximation and continuity correction probability that at most 165 transmission errors occur

=P(X<=165)=P(Z<(165.5-150)/11.29)=P(Z<1.37)=0.9147

b)

here let number of error in first and second message are X1 and X2

hence expected difference in errors =E(X1-X2)=E(X1)-E(X2)=150-150=0

and std deviation =SD(X1-X2)=(11.29²+11.26²)^1/2 =15.969

hence P(number of errors are within 60)=P(-60<X<60)=P(-59.5/15.969<Z<59.5/15.969)

=P(-3.73<Z<3.73)=0.9999-0.0001=0.9998