1. (5 points) A drawer contains 8 distinct pairs of socks (for a total of 16 socks). Reach into the drawer and pull out 4 socks at random. What is the probability that you have retrieved at least one pair?

Step 1 ：The total number of ways to draw 4 socks out of 16 is given by the combination formula \(C(n, r) = \frac{n!}{(n-r)!r!}\), where n is the total number of items, and r is the number of items to choose. In this case, n=16 and r=4, so the total number of ways is 1820.

Step 2 ：The total number of ways to draw 4 socks without getting a pair is a bit trickier. We can think of this as first choosing 4 different pairs (out of the 8 available), and then choosing one sock from each of these pairs. The number of ways to choose 4 pairs out of 8 is given by the combination formula with n=8 and r=4. For each of these choices, there are \(2^4\) ways to choose one sock from each pair (since each pair has 2 socks). So the total number of ways to draw 4 socks without a pair is 1120.

Step 3 ：The probability of drawing 4 socks without getting a pair is then the number of ways to draw 4 socks without a pair divided by the total number of ways to draw 4 socks, which is approximately 0.615.

Step 4 ：The probability of getting at least one pair is 1 minus this probability, which is approximately 0.385.

Step 5 ：Final Answer: The probability that you have retrieved at least one pair is approximately \(\boxed{0.385}\).