Question

jar has
13
13 marbles and
6
6 of them are green. You pull 3 marbles without replacement. What is the probability that all are green? Write your answer as a fraction. You do not need to simplify.

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Answer 1

To find the probability that all three marbles drawn from the jar are green, we can use the concept of conditional probability.

1. **Total number of marbles**: 13
2. **Total number of green marbles**: 6
3. **Number of marbles drawn**: 3

We need to consider the probability of drawing 3 green marbles in succession without replacement.

- **Probability of drawing the first green marble**:
\[
P(\text{1st green}) = \frac{6}{13}
\]

- **Probability of drawing the second green marble (after drawing the first)**:
After we draw one green marble, we have 5 green marbles left and 12 total marbles remaining. Thus,
\[
P(\text{2nd green | 1st green}) = \frac{5}{12}
\]

- **Probability of drawing the third green marble (after drawing the first and second)**:
After drawing two green marbles, we now have 4 green marbles left and 11 total marbles remaining. Thus,
\[
P(\text{3rd green | 1st and 2nd green}) = \frac{4}{11}
\]

Now, to find the total probability that all three marbles drawn are green, we multiply these probabilities together:

\[
P(\text{All green}) = P(\text{1st green}) \times P(\text{2nd green | 1st green}) \times P(\text{3rd green | 1st and 2nd green})
\]

Substituting the values:

\[
P(\text{All green}) = \frac{6}{13} \times \frac{5}{12} \times \frac{4}{11}
\]

Now, calculate the product:

\[
P(\text{All green}) = \frac{6 \times 5 \times 4}{13 \times 12 \times 11} = \frac{120}{1716}
\]

So, the probability that all three marbles drawn are green is:

\[
\frac{120}{1716}
\]

This is the final answer in a fraction form.