Understanding the Probability of Selecting Two Blue Socks

Let's dive into understanding the probability of selecting two blue socks from a box containing a mix of black, blue, and red socks. This article will guide you through each step, ensuring you can calculate this probability effectively.

Step 1: Determine the Total Number of Socks

Imagine a box containing 5 black socks, 4 blue socks, and 3 red socks. First, let's enumerate the total number of socks in the box.

Total Number of Socks

Black socks: 5 Blue socks: 4 Red socks: 3 Total socks: 5 4 3 12

Step 2: Calculate the Number of Ways to Choose 2 Blue Socks

Next, we need to determine the number of ways to choose 2 blue socks from the 4 blue socks available. We will use the combination formula for this calculation.

Combination Formula

The combination formula is given by:

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

Where ( n ) is the total number of items to choose from, and ( k ) is the number of items to choose.

In this case:
( n 4 ) (total blue socks)
( k 2 ) (number of blue socks to choose)

Let's calculate:

[ binom{4}{2} frac{4!}{2!(4-2)!} frac{4 times 3}{2 times 1} 6 ]

Step 3: Calculate the Total Number of Ways to Choose Any 2 Socks from 12

Now, we need to determine the total number of combinations when choosing any 2 socks from the 12 available socks.

Again, we use the combination formula:

[ binom{12}{2} frac{12!}{2!(12-2)!} frac{12 times 11}{2 times 1} 66 ]

Step 4: Calculate the Probability of Choosing Two Blue Socks

Finally, we can calculate the probability by dividing the number of favorable outcomes (choosing 2 blue socks) by the total number of possible outcomes (choosing any 2 socks).

Probability ( P ) is given by:

[ P(2 text{ blue socks}) frac{text{Number of ways to choose 2 blue socks}}{text{Total ways to choose 2 socks}} frac{6}{66} frac{1}{11} ]

Therefore, the probability of choosing two blue socks is ( frac{1}{11} ).

Alternative Method: Multiplication of Probabilities

Alternatively, we can approach the problem by considering the probabilities of two sequential events:

The probability that the first sock picked is blue is ( frac{4}{12} ) 33.33% ). Given that the first sock picked is blue, the probability that the second sock picked is also blue is ( frac{3}{11} ) 27.27% ).

The probability of both events occurring is the product of their individual probabilities:

[ frac{4}{12} times frac{3}{11} frac{1}{3} times frac{3}{11} frac{1}{11} ]

This confirms our previous calculation: the probability of selecting two blue socks is ( frac{1}{11} ) ).

Understanding how to calculate probabilities, such as this one, is crucial in various fields, including statistics and combinatorics. Whether you use combinatorial methods or the multiplication of probabilities, both approaches yield the same result.

Keywords: probability, combinatorics, sock selection