A Sequence of zero and ones (binary digits) is being transmitted through a commu

Question

A Sequence of zero and ones (binary digits) is being transmitted through a communication channel. Sometimes the channel mistakenly changes a 1 to a 0, or a 0 to a 1. As each bit is transmitted, an error occurs with probability p = 0.002. Assume that these errors occur independently.
A. The number of errors that will occur in a sequence of transmitted digits is a random variable with a binomial distribution. Explain why the binomial is the right distribution.
B. What is the probability that the sequence 1001 will be transmitted as 1111?

Explanation / Answer

A. A distribution is characterized as binomial when there are only two possible outcomes for each trail, there are a fixed number of trials, the outcomes are independent of each other, and the probability of success remains the same for each trial. In this case, all of these requirements are met. B. 1001 has 4 digits. There probability that any individual digit will be wrong is .002. Therefore the probability that they will be transmitted correctly is .998. The probability that the first digit is transmitted correctly: .998 The probability that the second digit is transmitted incorrectly: .002 The probability that the third digit is transmitted incorrectly: .002 The probability that the fourth digit is transmitted correctly: .998 The probability of getting right and wrong and wrong and right = .998 x .002 x .002 x .998 Therefore the probability of getting a 1111 is 3.98 x 10^-6

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