Question

Consider the problem of sending a binary message, 0 and 1, through a signal channel consisting of several stages, where transmission through each stage is subject to a fixed probability of error σ, denoted as 0. Suppose X ₀ is the signal that is sent and let Xₙ be the signal that is received at the nth stage. Assume that {X ₙ} is a Markov chain with transition probabilities P ₀₀ = P ₁₁ = 1- σ,and P ₀₁ = P ₁₀= σ, where 0 < σ < 1

Determine the probability that no error occurs up to stage n = 4.

Answer 1

The probability that no error occurs up to stage n = 4 is calculated as (1-σ)^4, given the transition probabilities and the Markov chain nature of the binary signal transmission system.

Step-by-step explanation:

The student has asked how to determine the probability that no error occurs up to stage n = 4 in a binary signal transmission which is a Markov chain with given transition probabilities. Since the states are binary (either 0 or 1), and there is a fixed probability σ of error at each stage, we understand that the system is subject to binomial conditions. To find the probability of no error up to stage 4, we need to compute the probability of the signal being transmitted correctly through each stage.

To achieve this without error up to the 4th stage, the signal needs to be transmitted correctly at each of the stages, which can be represented as (1-σ)^4 since the probability of correct transmission at each stage is 1-σ. Therefore, the probability of no error occurring up to stage n = 4 is:

P(No Error up to stage 4) = (1-σ)^4