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December 23rd, 2015, 09:11 AM  #1 
Senior Member Joined: Oct 2013 From: New York, USA Posts: 575 Thanks: 79  Probability With Marbles
There are 11 green marbles, 4 red marbles, and 5 blue marbles. You select 4 marbles without replacement. What are these probabilities? 1. 2 green marbles and 2 red marbles 2. 1 green marble and 3 red marbles I'm also interested in the probability of 4 red marbles and the probability of at least 1 blue marble, but I can do those myself. 
December 23rd, 2015, 09:41 AM  #2 
Senior Member Joined: Mar 2011 From: Chicago, IL Posts: 214 Thanks: 77 
1. $\displaystyle \frac{\binom{11}{2}\cdot\binom{4}{2}}{\binom{11+4+ 5}{4}}$

December 23rd, 2015, 07:28 PM  #3 
Member Joined: Dec 2015 From: Down Under Posts: 32 Thanks: 3 
Hi skaa, I am unfamiliar with this notation. Could you please give a brief explanation? If I wanted to solve this problem, I would've had to go through the arduous process of adding the products of each possible sequence that qualifies. 
December 24th, 2015, 06:51 AM  #4 
Senior Member Joined: Apr 2015 From: Planet Earth Posts: 121 Thanks: 23 
$\displaystyle \binom{N}{M}$ is called a combination. It means the number of ways you can take M marbles from a bag of N without replacing them. It evaluates to N!/M!/(NM!). So, for example, if you take two marbles from a bag containing four labeled {A,B,C,D}; you can get {A,B}, {A,C}, {A,D}, {B,C}, {B,D} or {C,D}. That's six combinations, or 4!/2!/(42)! = (4*3*2*1)/(2*1)/(2*1)=6. If you take 4 marbles from 11+4+5=20, there are $\displaystyle \binom{20}{4}$=4845 ways to do it. To get 2 of the 11 green, and 2 of the 4 red, there are $\displaystyle \binom{11}{2}*\binom{4}{2}$=330 ways. The chances of getting that combination are thus 330/4845~=.0681. The second one can be done the same way. 
December 24th, 2015, 07:13 AM  #5 
Member Joined: Dec 2015 From: Down Under Posts: 32 Thanks: 3 
I see! I had just been using the equivalent formula: N! / ((N  M)!(M!)) Glad there is a simpler way to write it! Haha. Makes the process understandable (: Thank you 

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