2. (5 points) Consider the experiment of drawing 4 cards at random and without replacement from a well-shuffled deck of 52 cards. What is the probability that you draw exactly 2 pairs? Using the notation of the class, this is represented by x, x, y, y.

Step 1 ：Consider the experiment of drawing 4 cards at random and without replacement from a well-shuffled deck of 52 cards. We want to find the probability that we draw exactly 2 pairs.

Step 2 ：First, calculate the total number of ways to draw 4 cards from a deck of 52 cards. This is given by the combination formula \(C(n, k) = \frac{n!}{k!(n-k)!}\), where n is the total number of items, k is the number of items to choose, and ! denotes factorial. In this case, n = 52 and k = 4. So, the total number of ways to draw 4 cards is \(C(52, 4) = 270725\).

Step 3 ：Next, calculate the number of ways to draw exactly 2 pairs. We first choose 2 ranks out of 13 for the pairs, which can be done in \(C(13, 2)\) ways. For each chosen rank, we then choose 2 cards out of 4, which can be done in \(C(4, 2)\) ways. Therefore, the total number of ways to draw exactly 2 pairs is \(C(13, 2) * C(4, 2) * C(4, 2) = 2808\).

Step 4 ：The probability is then given by the ratio of the number of ways to draw exactly 2 pairs to the total number of ways to draw 4 cards. So, the probability is \(\frac{2808}{270725} = 0.010372148859543818\).

Step 5 ：Final Answer: The probability that you draw exactly 2 pairs is approximately \(\boxed{0.0104}\).