# Non-homework probability question.

#### mclark

##### New Member
Hello,

I have a question related to an online game I play.

Suppose you have 100 marbles in a bag. 70 are red, 15 are yellow, 10 green, and 5 blue.

You pull out 15 marbles. What is the probability of the final result as 12 red, with the other 3 any other color?

I tried 70/100*69/99*68/98 etc, and got a really small number.

Or I turn it around and figure well I start out with 70% red, ideally I would get 70% red in my result or 10.5 red marbles. So an extra 1.5 red marbles seems not too unlikely, with a SWAG of 30% chance.

Do the results change if you have 10,000 marbles, with the same distribution, ie 70% red, 15% yellow and so on?

My background on probability is low. I took Statistics 101 9 years ago and am good at math. I am not currently familiar with probability jargon.

Thanks for any insight.

PS The game is World of Tanks, and if you're familiar with the game, you'll guess why I'm asking. I can explain more if you're really interested. The game is addictive, so use caution if you start to play.

#### BGM

##### TS Contributor
So you have 70% red, 30% non-red, and you randomly sample 15 of them.

The number of red follows a hypergeometric distribution.

The probability of getting exactly 12 red is

$$\frac {\displaystyle \binom {0.7N} {12} \binom {0.3N} {3}} {\displaystyle \binom {N} {15}}$$

where $$N$$ is a positive multiple of 10, representing the population size.

You can plug in $$N = 100$$ or $$N = 10000$$ to see the various results.

#### mclark

##### New Member
Thank you! The binomial coefficient calculator I found online gives me about 17% for both N=100 and N=500. It doesn't let N go any higher than that.

#### rogojel

##### TS Contributor
hi,
for a large N the probability should be very close to the simple binomial B(15,12) , because the difference between sampling without teplacement and with replacement will become negligible.

regards
rogojel