Question

A box has 3 white marbles, 5 green marbles, and 2 yellow marbles. Two marbles are randomly selected without replacement. What is the probability of selecting two green marbles?

Write the answer in the simplest form of a fraction. (For example: 1/2 )
*
1 point

Answer 1

To find the probability of selecting two green marbles from the box, we first need to determine the total number of marbles, as well as the total number of ways to choose two green marbles.

1. **Total number of marbles:**
- 3 white marbles
- 5 green marbles
- 2 yellow marbles
- Total = 3 + 5 + 2 = 10 marbles

2. **Calculating the total number of ways to choose 2 marbles from 10:**
The number of ways to choose 2 marbles from 10 is given by the combination formula \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \):
\[
\binom{10}{2} = \frac{10!}{2!(10-2)!} = \frac{10 \times 9}{2 \times 1} = 45
\]

3. **Calculating the number of ways to choose 2 green marbles from 5:**
Similarly, the number of ways to choose 2 green marbles from 5 is:
\[
\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10
\]

4. **Calculating the probability of selecting 2 green marbles:**
The probability is the ratio of the number of favorable outcomes to the total number of outcomes:
\[
P(\text{2 green marbles}) = \frac{\text{Number of ways to choose 2 green}}{\text{Total ways to choose 2 marbles}} = \frac{10}{45}
\]

5. **Simplifying the fraction:**
\[
\frac{10}{45} = \frac{2}{9}
\]

Thus, the probability of selecting two green marbles is \(\boxed{\frac{2}{9}}\).