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Lets say I have red, green, yellow and blue balls in an infinitely large bag containing an infinite number of balls in total, where the amounts of each ball are described by a percentage (as as an irrational number).

Lets say that the percentages are as follows:

(A) red    55.0%
(B) green  30.0%
(C) yellow 10.5%
(D) blue    4.5%

Now lets say that I can select three balls from the bag with replacement, using combinatorial methods, the possible combinations can be determined to be as follows:

 A B C D
 0 0 0 3
 0 0 1 2
 0 0 2 1
 0 0 3 0
 0 1 0 2
 0 1 1 1
 0 1 2 0
 0 2 0 1
 0 2 1 0
 0 3 0 0
 1 0 0 2
 1 0 1 1
 1 0 2 0
 1 1 0 1
 1 1 1 0
 1 2 0 0
 2 0 0 1
 2 0 1 0
 2 1 0 0
 3 0 0 0

How can I work out the probability of each instance (row) occurring?

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  • $\begingroup$ Look say in Wikipedia for multinomial distribution. $\endgroup$ – André Nicolas Jun 25 '15 at 6:11
  • $\begingroup$ Great. Thats it, Cheers. $\endgroup$ – ADP Jun 25 '15 at 6:13
  • $\begingroup$ @Andre, That solved it, if you want to write up a brief answer, I'll check it as the solution... $\endgroup$ – ADP Jun 25 '15 at 6:18
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We are effectively sampling with replacement. The relevant distribution is the multinomial distribution.

Let our probabilities be $p_1,p_2,p_3,p_4$. The probability that a sample of size $n$ has $k_1$ red, $k_2$ green, $k_3$ yellow, and $k_4$ blue is $$\binom{n}{k_1,k_2,k_3,k_4}p_1^{k_1}p_2^{k_2}p_3^{k_3}p_4^{k_4},$$ where $\binom{n}{k_1,k_2,k_3,k_4}=\frac{n!}{k_1!k_2!k_3!k_4!}$.

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