Probability of Drawing at Least One Matching Pair of Socks

Imagine you are standing in front of a drawer that contains 16 socks, split into 8 pairs. Your goal is to randomly and without replacement choose 6 socks from this drawer. The question is: what is the probability that among these 6 socks, you will find at least one matching pair?

Using Complementary Probability

To solve this problem, we will use the complementary probability method. This involves calculating the probability of the complementary event (drawing 6 socks with no matching pairs) and then subtracting this from 1. This way, we can find the probability of the original event, which is having at least one matching pair.

Step 1: Total Ways to Choose 6 Socks

First, we need to determine the total number of ways to choose 6 socks from the 16 socks available. This is a classic combination problem, where order does not matter. We use the combination formula:

$$ binom{n}{k} frac{n!}{k!cdot (n-k)!} $$

Here, ( n ) is the total number of socks (16) and ( k ) is the number of socks to choose (6). Plugging these values into the formula gives us:

$$ binom{16}{6} frac{16!}{6!cdot 10!} 8008 $$

This means there are 8008 different ways to choose 6 socks from 16.

Step 2: Ways to Choose 6 Socks Without a Matching Pair

Next, we calculate the number of ways to choose 6 socks without any matching pairs. To ensure no matches, we must select one sock from each of 6 different pairs. Here's the step-by-step calculation:

Step 2a: Choosing 6 Pairs from 8

We can select 6 pairs from the 8 available pairs in 28 different ways. This is calculated using the combination formula:

$$ binom{8}{6} binom{8}{2} 28 $$

There are 28 ways to choose 6 pairs from 8.

Step 2b: Choosing One Sock from Each Pair

Once we have selected the 6 pairs, we can choose one sock from each pair. For each of the 6 pairs, we have 2 choices (either the left or the right sock). Thus, the number of ways to choose one sock from each of the 6 pairs is:

$$ 2^6 64 $$

This means there are 64 ways to choose one sock from each of the 6 pairs.

Total Number of Ways to Choose 6 Socks Without a Matching Pair

The total number of ways to choose 6 socks without a matching pair is the product of the ways to choose the pairs and the ways to choose one sock from each pair:

$$ 28 times 64 1792 $$

Step 3: Probability of Choosing 6 Socks Without a Matching Pair

Now, we can calculate the probability of drawing 6 socks without any matching pairs:

$$ P(text{no matching pair}) frac{1792}{8008} $$

Calculating this gives:

$$ P(text{no matching pair}) approx 0.223 $$

Step 4: Probability of At Least One Matching Pair

The final step is to find the probability of having at least one matching pair. This is the complement of the probability of having no matching pairs:

$$ P(text{at least one matching pair}) 1 - P(text{no matching pair}) 1 - frac{1792}{8008} $$

Calculating this:

$$ P(text{at least one matching pair}) approx 1 - 0.223 approx 0.777 $$

Final Answer

Therefore, the probability that there is at least one matching pair among the six socks is approximately:

$$ boxed{0.777} $$

Simplifying the Problem

When we simplify the probability problem to its core, we are essentially asking how likely it is to pick 6 socks without getting a pair, given that there are 8 pairs available. Using combinatorial methods, we can calculate this probability accurately and provide a clear answer.

Key Takeaways

The total number of ways to choose 6 socks out of 16 is 8008. The number of ways to choose 6 socks without a matching pair is 1792. The probability of drawing 6 socks without a matching pair is approximately 0.223. The probability of drawing at least one matching pair is the complement of the above, approximately 0.777.

Application in Real Life

This type of probability problem is not just a theoretical exercise. It can be applied in various real-life scenarios, such as:

Quality control in manufacturing where misclassified items are to be avoided. In games of chance, where understanding probabilities can provide strategic advantages. Efficient data handling in computer science and information systems to avoid collisions in hash tables.

Conclusion

By understanding the probability calculations and methods, you can tackle similar problems in various fields and make more informed decisions. Using the complementary probability method is a powerful technique that simplifies complex problems.