A transmitter wants to convey a signal which can be RED with probability q or GR

Question

A transmitter wants to convey a signal which can be RED with probability q or GREEN, with a probability of 1 - q, to a receiver. The transmitter has an input signal X which can be chosen to be 0 or 1. The receiver receives a signal Y as follows. Pr(Y = 0|X = 0) = Pr(Y = 1|X = 1) = p Pr(Y = 0|X = 1) = Pr(Y = 1|X = 0) = 1 - p The transmitter uses the following strategy: If the signal is RED, the transmitter sets X = 0. If the signal is GREEN, the transmitter sets X = 1. The receiver uses the following strategy: If it receives Y = 0, it declares the color as RED. If it receives Y = 1, it declares the color as GREEN. Evaluate the probability that the receiver declares the wrong color.

Explanation / Answer

The receiver declares the wrong color when actually Y =1 but it should be Y = 0, or say, X = 0 and when Y = 0 but it should be Y = 1, or say, X = 1.

Hence the probability that the receiver declares the wrong color = P(Y = 1, X = 0) + P(Y = 0 , X = 1)

= P(Y = 1 | X = 0) P(X = 0) + P(Y = 0 | X = 1) P(X = 1) = (1-p) q + (1-p) (1 - q) = (1 - p). (Ans).

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