To find the probability that all three marbles drawn from the jar are green, we can use the concept of conditional probability.
- Total number of marbles: 13
- Total number of green marbles: 6
- Number of marbles drawn: 3
We need to consider the probability of drawing 3 green marbles in succession without replacement.
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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.