Question

A drawer contains loose socks. There are 2 blue and 4 black socks in the drawers. What is the probability that you choose a sock without looking, then choose a second sock (keeping the first in your hand) without looking and end up with a pair of black socks?

Answer 1

To find the probability of drawing a pair of black socks when selecting two socks from a drawer that contains 2 blue socks and 4 black socks, we first determine the total number of socks and the total ways to choose two socks.

### Step 1: Calculate Total Socks
- Blue socks: 2
- Black socks: 4
Total socks = 2 + 4 = 6

### Step 2: Choose Two Socks
We can choose the first sock in several ways. We need to calculate the probability of specific outcomes:

1. The total number of ways to select 2 socks from 6 is given by the combination formula \( \binom{n}{k} \), where \( n \) is the total number of items to choose from, and \( k \) is the number of items to choose.

\[
\text{Total combinations to choose 2 socks} = \binom{6}{2} = \frac{6 \times 5}{2 \times 1} = 15
\]

### Step 3: Favorable Outcomes (Both Black Socks)
Next, we calculate the number of ways to choose 2 black socks from the 4 available:

\[
\text{Combinations to choose 2 black socks} = \binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6
\]

### Step 4: Calculate Probability
Now we can calculate the probability of selecting a pair of black socks.

\[
\text{Probability of choosing 2 black socks} = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{6}{15} = \frac{2}{5}
\]

Thus, the probability of choosing a pair of black socks is

\[
\boxed{\frac{2}{5}}.
\]