# Thread: drawing marbles from a bag

1. ## drawing marbles from a bag

if there are 20 white marbles and 32 black marbles in a bag
what is the probability of drawing 20 marbles where 10 are white and 10 are black ?
is there a formula ?

2. Originally Posted by addison
if there are 20 white marbles and 32 black marbles in a bag
what is the probability of drawing 20 marbles where 10 are white and 10 are black ?
is there a formula ?
Yes. Read this: Hypergeometric distribution - Wikipedia, the free encyclopedia.

In your case: N = 52, n = 20, m = 20, k = 10.

There are 20 white marbles and 32 black marbles in a bag.
What is the probability of drawing 20 marbles where 10 are white and 10 are black?
There are: . ${52\choose20}$ possible outcomes.

To draw 10 white marbles, there are: . ${20\choose10}$ ways.

To draw 10 black marbles, there are: . ${32\choose10}$ ways.

. . Hence, there are:\ . ${20\choose10}{32\choose10}$ ways to get 10 white and 10 black marbles.

Therefore: . $\boxed{P(\text{10 white, 10 black}) \;=\;\frac{{20\choose10}{32\choose10}}{{52\choose2 0}}}$