Question

Find the probability that two socks selected at random from a drawer containing 10 black socks and 6 white socks will both be black.

Question 13 options:

62.5%


50%


39.1%


37.5%

Answer 1

To find the probability that both socks selected at random are black, we can use the concept of combinations.

1. First, determine the total number of socks:
- Black socks = 10
- White socks = 6
- Total socks = 10 + 6 = 16

2. The number of ways to choose 2 black socks from 10 black socks can be calculated using the combination formula \( C(n, k) = \frac{n!}{k!(n-k)!} \), where \( n \) is the total number of items to choose from and \( k \) is the number of items to choose.

\[
C(10, 2) = \frac{10!}{2!(10-2)!} = \frac{10 \times 9}{2 \times 1} = 45
\]

3. The total number of ways to choose 2 socks from the total of 16 socks is:

\[
C(16, 2) = \frac{16!}{2!(16-2)!} = \frac{16 \times 15}{2 \times 1} = 120
\]

4. Now, calculate the probability that both selected socks are black:
\[
P(\text{both socks black}) = \frac{\text{Number of ways to choose 2 black socks}}{\text{Total ways to choose 2 socks}} = \frac{45}{120}
\]

5. Simplifying the fraction:
\[
P(\text{both socks black}) = \frac{45}{120} = \frac{3}{8}
\]

6. Convert the fraction to a percentage:
\[
\frac{3}{8} = 0.375 = 37.5\%
\]

Thus, the probability that both socks selected will be black is **37.5%**.

Therefore, the correct option is:
**37.5%**