A digital communications system handles binary signals ones

A digital communications system handles binary signals, ones and zeros. Because of noise, some bits arrive in error with probability p. Assume that each bit is statistically independent. Find the probability that four successive symbols arrive without error four successive symbols arrive with exactly one error four successive symbols arrive with two or more errors four successive symbols arrive with two or fewer errors

Solution

a)

P(4 successive errors) = P(error)^4 = p^4 [answer]

b)

Using binomial distribution,

P(exactly 1 error) = 4C1 p^1 (1-p)^3 = 4 p^1 (1-p)^3 [answer]

c)

P(2 or more errors) = 1 - P(0 error) - P(1 error) = 1 - (1-p)^4 - 4 p^1 (1-p)^3 [answer]

d)

P(2 or fewer errors) = P(0) + P(1) +P(2)

= (1-p)^4 + 4 p^1 (1-p)^3 + 4C2 p^2 (1-p)^2

= (1-p)^4 + 4 p^1 (1-p)^3 + 6 p^2 (1-p)^2 [answer]