A digital communication channel has an error rate of 107 wh

A digital communication channel has an error rate of 10?7 , which may be taken to mean that the probability of any bit (a one or a zero) arriving incorrectly at the destination is 10?7 . What is the probability that a transmission of a million bits occurs without a single error? It is reasonable to assume that the transmission of a given bit (incorrectly or correctly) does not affect the transmission of other bits. State clearly the random variable and the probability distribution you are using and the values of the parameters associated with that distribution.

Solution

Let X: Number of bits arriving incorrectly(with error) at the destination

Here n = Number bits tobe transmitted = 10⁶ = 1000000, p = the probability of any bit (a one or a zero) arriving incorrectly at the destination = 10^-7.

It is reasonable to assume that the transmission of a given bit (incorrectly or correctly) does not affect the transmission of other bits that is transmission of each bit is independent of the other and n is finite so we can use binomial distribution. X follows binomial with n=10⁶ and p = 10^-7 .

The probability that a transmission of a million bits occurs without a single error = P[ X =0]

                                                                                                                 = ^1000000C₀ (10^-7)⁰(1 - 10^-7)^1000000-0

                                                                                                                  = 0.9048

The probability that a transmission of a million bits occurs without a single error = 0.9048.