Two red marbles from a small bag

Question

A cloth bag holds 3 red marbles and 2 green marbles. You reach in and pull out 2 marbles at once, without looking. What is the probability that both marbles are red?

Accepted Answer

Think of drawing one marble, then the second, and multiply the chances. The first marble is red with probability 	frac{3}{5}. Given that, four marbles remain, 2 of them red, so the second is red with probability 	frac{2}{4}: P(	ext{both red}) = \frac{3}{5} 	imes \frac{2}{4} = \frac{6}{20} = \frac{3}{10}. A counting check gives the same answer: there are \binom{5}{2}=10 equally likely pairs, and \binom{3}{2}=3 of them are all-red, so P = 	frac{3}{10}. Takeaway. For draws without replacement, chain the conditional probabilities: shrink the pool after each pick. Counting pairs with combinat…

Two red marbles from a small bag

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