Question


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.

Answers

Answered by GPT 3.5
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:
$\dfrac{4}{11} \times \dfrac{3}{10} \times \dfrac{2}{9} = \dfrac{4 \times 3 \times 2}{11 \times 10 \times 9} = \dfrac{24}{990} = \boxed{\dfrac{4}{165}}$.
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