The random variable $$\(X\)$$ takes each of the values $$\(1,2,3,4\)$$ with probability $$\(\frac{1}{4}\)$$. Two independent values of $$\(X\)$$ are chosen at random. If the two values of $$\(X\)$$ are the same, the random variable $$\(Y\)$$ takes that value. Otherwise, the value of $$\(Y\)$$ is the larger value of $$\(X\)$$ minus the smaller value of $$\(X\)$$. Find the probability that $$\(Y=2\)$$ given that $$\(Y\)$$ is even. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

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