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What is the probability she gets exactly 3 that are the same color?

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A vase ccontains 16 yellow, 12 green, 9 purple and 3 red marbles. if lily randomly reaches in the vase and randomly pulls out 5 marbles without replacement, what is the probability she gets exactly she gets 3 that are the same color

Apr 22nd, 2015

THREE YELLOWS: 
From the 16 yellows choose 3 --> 16C3 = 560 
From the 24 non-yellows choose 2 --> 24C2 = 276 
560 x 276 = 154,560 ways 

THREE GREENS 
From the 12 greens choose 3 --> 12C3 = 220 
From the 28 non-greens choose 2 --> 28C2 = 378 
220 x 378 = 83,160 ways 

THREE PURPLES 
From the 9 purples choose 3 --> 9C3 = 84 
From the 31 non-purples choose 2 --> 31C2 = 465 
84 x 465 = 39,060 ways 

THREE REDS 
From the 3 reds choose 3 --> 3C3 = 1 
From the 37 non-reds choose 2 --> 37C2 = 666 
1 x 666 = 666 ways 

THREE OF ANY COLOR 
154,560 + 83,160 + 39,060 + 666 = 277,446 ways 

Now count the number of ways to get *any* 5 marbles. 

ANY 5 MARBLES 
From the 40 marbles choose 5 --> 658,008 ways 

Finally compute the probability: 
P(exactly 3 of the same color) = 277,446 / 658,008 

That can be reduced to: 
3557 / 8436 
 ~42.16%

Apr 22nd, 2015

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