Sure, I’ll guide you through the solution process step-by-step.
### Step-by-Step Solution:
1. Define Variables and Probabilities:
- The input, [tex]\( x \)[/tex], to the binary communication system takes two values: [tex]\( x = 0 \)[/tex] or [tex]\( x = 1 \)[/tex].
- The probabilities of these inputs are:
[tex]\[
P(X=0) = \frac{1}{4}, \quad P(X=1) = \frac{3}{4}
\][/tex]
2. Noise Probabilities:
- Define the probabilities that the output differs from the input due to noise as follows:
[tex]\[
P(Y=1 \mid X=0) = 0.1, \quad P(Y=0 \mid X=1) = 0.2
\][/tex]
3. Correct Transmission Probabilities:
- For [tex]\( x = 0 \)[/tex] to be correctly transmitted, the output should also be 0. This probability is:
[tex]\[
P(Y=0 \mid X=0) = 1 - P(Y=1 \mid X=0) = 1 - 0.1 = 0.9
\][/tex]
- For [tex]\( x = 1 \)[/tex] to be correctly transmitted, the output should also be 1. This probability is:
[tex]\[
P(Y=1 \mid X=1) = 1 - P(Y=0 \mid X=1) = 1 - 0.2 = 0.8
\][/tex]
4. Calculate Joint Probabilities for Correct Transmission:
- For [tex]\( x = 0 \)[/tex] and [tex]\( y = 0 \)[/tex]:
[tex]\[
P(X=0 \text{ and } Y=0) = P(Y=0 \mid X=0) \times P(X=0) = 0.9 \times \frac{1}{4} = 0.225
\][/tex]
- For [tex]\( x = 1 \)[/tex] and [tex]\( y = 1 \)[/tex]:
[tex]\[
P(X=1 \text{ and } Y=1) = P(Y=1 \mid X=1) \times P(X=1) = 0.8 \times \frac{3}{4} = 0.6
\][/tex]
5. Total Probability of Correct Transmission:
- Summing the joint probabilities calculated:
[tex]\[
P(\text{correct transmission}) = P(X=0 \text{ and } Y=0) + P(X=1 \text{ and } Y=1) = 0.225 + 0.6 = 0.825
\][/tex]
6. Probability of Incorrect Transmission:
- Since the output is either correct or incorrect, the probability of incorrect transmission is:
[tex]\[
P(\text{incorrect transmission}) = 1 - P(\text{correct transmission}) = 1 - 0.825 = 0.175
\][/tex]
### Final Results:
- Probability that [tex]\( x = 0 \)[/tex] is correctly transmitted ([tex]\( P(X=0 \text{ and } Y=0) \)[/tex]): [tex]\( 0.225 \)[/tex]
- Probability that [tex]\( x = 1 \)[/tex] is correctly transmitted ([tex]\( P(X=1 \text{ and } Y=1) \)[/tex]): [tex]\( 0.6 \)[/tex]
- Total probability of correct transmission: [tex]\( 0.825 \)[/tex]
- Total probability of incorrect transmission: [tex]\( 0.175 \)[/tex]
These calculations and results provide a complete solution to the question.
