What is the average probability of error with two symbols 0 and 1 with source probability 0.2, 0.8 respectively transmitted over a binary asymmetric channel as given below:

Explanation:

Average Probability of Error in a Binary Asymmetric Channel:

To solve the given problem, we need to calculate the average probability of error when transmitting two symbols, 0 and 1, over a binary asymmetric channel. The channel is characterized by different probabilities of error when transmitting 0 or 1. Let’s begin by understanding the problem and calculating the required values step-by-step.

Definitions and Notations:

• Let P(0) and P(1) be the source probabilities of symbols 0 and 1, respectively.
• Let P(error | 0) and P(error | 1) be the conditional probabilities of error when transmitting 0 and 1, respectively.
• The average probability of error, denoted as P(error), can be calculated using the formula:
  
  P(error) = P(0) × P(error | 0) + P(1) × P(error | 1)

Given Data:

• Source probabilities: P(0) = 0.2, P(1) = 0.8
• Conditional probabilities of error: P(error | 0) = 0.1, P(error | 1) = 0.4

Step-by-Step Calculation:

P(error) = P(0) × P(error | 0) + P(1) × P(error | 1)

= (0.2 × 0.1) + (0.8 × 0.4)

(0.2 × 0.1) = 0.02

(0.8 × 0.4) = 0.32

P(error) = 0.02 + 0.32 = 0.38

1. Substitute the given values into the formula for P(error):
2. Perform the calculations:
3. Add the results to find the average probability of error:

Final Answer: The average probability of error is 0.38.