Question

A jar has 10 marbles and 5 of them are green. You pull 2 marbles without replacement. What is the probability that both are green? Write your answer as a fraction. You do not need to simplify.

Answer 1

To find the probability that both marbles drawn are green when pulling 2 marbles from a jar containing 10 marbles (5 of which are green), we can approach this using conditional probability.

1. **Determine the total number of ways to choose 2 marbles from 10 marbles.**
This can be calculated using the binomial coefficient:

\[
\binom{10}{2} = \frac{10 \times 9}{2 \times 1} = 45
\]

2. **Determine the number of ways to choose 2 green marbles from the 5 green marbles.**
This is also calculated using the binomial coefficient:

\[
\binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10
\]

3. **Calculate the probability that both marbles drawn are green.**
The probability can be calculated as the ratio of the number of favorable outcomes (choosing 2 green marbles) to the total outcomes (choosing any 2 marbles):

\[
P(\text{both are green}) = \frac{\text{Number of ways to choose 2 green marbles}}{\text{Total number of ways to choose 2 marbles}} = \frac{10}{45}
\]

Thus, the probability that both marbles drawn are green is:

\[
\frac{10}{45}
\]

This is the final answer, and there is no need to simplify it. The probability that both marbles are green is:

\[
\frac{10}{45}
\]