Probability of Selecting Exactly One Complete Pair of Socks from a Drawer

Let’s embark on the journey of calculating the probability of a specific scenario: choosing exactly one complete pair of socks when randomly picking 6 out of 20 distinct pairs of socks. This problem combines combinatorial mathematics and probability theory to offer a profound insight into how these principles work together.

Understanding the Problem

The problem involves selecting 6 socks from a total of 40 distinct socks, where 20 of these socks are distinct pairs. We are interested in finding the probability of having exactly one complete pair among the selected 6 socks.

Step-by-Step Solution

To solve this, we will break the process into several steps, ensuring we cover every detail meticulously.

Step 1: Calculate the Total Number of Ways to Choose 6 Socks

The first step is to find the total number of ways to choose 6 socks from 40. This can be calculated using the combination formula:

[binom{n}{k} frac{n!}{k!(n-k)!}]

Substituting the values, we get:

[binom{40}{6} frac{40!}{6! cdot 34!}]

The factorial 40! / (6! * 34!) 38,276,640 ways to choose 6 socks from 40.

Step 2: Choose the Complete Pair and Additional Socks

To have exactly one complete pair, we proceed as follows:

Choose 1 Complete Pair of Socks

There are 20 pairs of socks, so the number of ways to choose 1 pair is simply:

20 ways.

Choose 4 Additional Socks

The 4 remaining socks must be chosen from the remaining 38 socks, which consist of 19 pairs. We need to ensure no additional pairs are selected, which means we can only choose 1 sock from each of the 4 different pairs.

First, choose 4 pairs from the remaining 19 pairs. The number of ways to do this is: [binom{19}{4} frac{19!}{4! cdot 15!} 4845] Second, choose 1 sock from each of the 4 pairs, and there are 2 ways to choose 1 from each pair. The total number of ways is: [2^4 16]

Step 3: Combine the Choices

Now, combine the number of ways to choose 1 complete pair and 4 additional socks:

[text{Favorable outcomes} 20 times binom{19}{4} times 2^4 20 times 4845 times 16 1,240,320]

Step 4: Calculate the Total Combinations

We have already calculated that the total number of ways to choose 6 socks from 40 as 38,276,640.

Step 5: Calculate the Favorable Outcomes

The numbers are already calculated as shown above:

[text{Favorable outcomes} 1,240,320]

Step 6: Calculate the Probability

Finally, the probability (P) of getting exactly one complete pair is:

[P frac{text{Favorable outcomes}}{text{Total outcomes}} frac{1,240,320}{38,276,640} approx 0.323]

This means the probability of selecting exactly one complete pair of socks when picking 6 randomly from a drawer containing 20 pairs of socks is approximately 0.323, or 32.3%.

Conclusion

The beauty of this calculation lies in its precision and the clear application of combinatorial mathematics to a real-world problem. Understanding such concepts not only enhances problem-solving skills but also deepens one's appreciation for the elegant simplicity of mathematics.