To determine if events X and Y are independent, we need to check if \(P(X \cap Y) = P(X) \times P(Y)\).

First, calculate \(P(X)\):

The possible outcomes for the total score to be 4 are (1,3), (2,2), and (3,1). Thus, there are 3 favorable outcomes.

Since there are 36 possible outcomes when a die is thrown twice, \(P(X) = \frac{3}{36}\).

Next, calculate \(P(Y)\):

The first score can be 2 or 5. For each of these, there are 6 possible outcomes for the second die. Thus, there are 12 favorable outcomes.

So, \(P(Y) = \frac{12}{36}\).

Now, calculate \(P(X \cap Y)\):

The outcomes that satisfy both X and Y are (2,2). Thus, there is 1 favorable outcome.

So, \(P(X \cap Y) = \frac{1}{36}\).

Check for independence:

\(P(X) \times P(Y) = \frac{3}{36} \times \frac{12}{36} = \frac{1}{36}\).

Since \(P(X \cap Y) = P(X) \times P(Y)\), events X and Y are independent.