A jar has
11
11 marbles and
4
4 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.
To find the probability that all three marbles are green, we can think of it as the probability of pulling a green marble in the first draw, then a green marble in the second draw, and finally a green marble in the third draw.
The probability of pulling a green marble in the first draw is $\dfrac{4}{11}$.
After the first marble is drawn, there are now 10 marbles left in the jar, with 3 green marbles remaining. So the probability of pulling a green marble in the second draw, without replacement, is $\dfrac{3}{10}$.
Similarly, after the second marble is drawn, there are 9 marbles left in the jar, with 2 green marbles remaining. So the probability of pulling a green marble in the third draw, without replacement, is $\dfrac{2}{9}$.
To find the probability that all three marbles are green, we multiply the probabilities of each individual draw:
$\dfrac{4}{11} \times \dfrac{3}{10} \times \dfrac{2}{9} = \dfrac{4 \times 3 \times 2}{11 \times 10 \times 9} = \dfrac{24}{990} = \boxed{\dfrac{4}{165}}$.