Question

A jar has
18 marbles and
7 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 from the jar are green, we can use the concept of conditional probability.

There are a total of 18 marbles, and out of those, 7 are green.

1. The probability that the first marble drawn is green:
\[
P(\text{First is green}) = \frac{7}{18}
\]

2. If the first marble drawn is green, there will be 6 green marbles left in the jar out of a total of 17 marbles (since one has been removed). The probability that the second marble drawn is green given that the first was green:
\[
P(\text{Second is green | First is green}) = \frac{6}{17}
\]

3. The overall probability that both marbles drawn are green is the product of the two probabilities:
\[
P(\text{Both are green}) = P(\text{First is green}) \times P(\text{Second is green | First is green}) = \frac{7}{18} \times \frac{6}{17}
\]

Calculating this:
\[
P(\text{Both are green}) = \frac{7 \times 6}{18 \times 17} = \frac{42}{306}
\]

Thus, the probability that both marbles drawn are green is:
\[
\frac{42}{306}
\]

This is the final answer, and you do not need to simplify it.