In a communication system, bit 0 is transmitted with probability 0.6 and bit 1 with 0.4. Due to channel noise, a transmitted 0 is received as 1 with probability 0.1, and a transmitted 1 is received as 0 with probability 0.2. If 1 is received, what is the probability that 1 was transmitted?
Answer and explanation
Correct answer: A. 16/19
- Step 1: Apply Bayes' theorem: P(T1 | R1) = P(R1 | T1) P(T1) / P(R1).
- Step 2: P(R1 | T1) = 1 - 0.2 = 0.8; P(T1) = 0.4. Numerator = 0.8 * 0.4 = 0.32.
- Step 3: P(R1 | T0) = 0.1; P(T0) = 0.6. Total P(R1) = 0.6 0.1 + 0.4 0.8 = 0.06 + 0.32 = 0.38.
- Step 4: P(T1 | R1) = 0.32 / 0.38 = 32 / 38 = 16/19.
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