(a) To find the probability distribution of Y, consider all possible pairs of X values:

• \((1,1)\), \((2,2)\), \((3,3)\), \((4,4)\): Y takes the value 1, 2, 3, 4 respectively.
• \((1,2)\), \((2,1)\): Y = 1.
• \((1,3)\), \((3,1)\): Y = 2.
• \((1,4)\), \((4,1)\): Y = 3.
• \((2,3)\), \((3,2)\): Y = 1.
• \((2,4)\), \((4,2)\): Y = 2.
• \((3,4)\), \((4,3)\): Y = 1.

Calculate probabilities:

• \(P(Y=1) = \frac{7}{16}\)
• \(P(Y=2) = \frac{5}{16}\)
• \(P(Y=3) = \frac{3}{16}\)
• \(P(Y=4) = \frac{1}{16}\)

(b) Given Y is even, possible values are 2 and 4. Calculate:

\(P(Y=2 \mid Y \text{ is even}) = \frac{P(Y=2)}{P(Y=2) + P(Y=4)} = \frac{\frac{5}{16}}{\frac{5}{16} + \frac{1}{16}} = \frac{5}{6}\)