# Probability of Drawing Balls from Infinite Bag

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?

• Look say in Wikipedia for multinomial distribution. – André Nicolas Jun 25 '15 at 6:11
• Great. Thats it, Cheers. – ADP Jun 25 '15 at 6:13
• @Andre, That solved it, if you want to write up a brief answer, I'll check it as the solution... – ADP Jun 25 '15 at 6:18

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 $\binom{n}{k_1,k_2,k_3,k_4}=\frac{n!}{k_1!k_2!k_3!k_4!}$.