probability with percentage independence There is 15% chance of choosing blue marble, 65% chance of choosing a purple marble, and 20% chance of choosing a black marble, there will be 18 total draws,
what is the probability that you will choose exactly 12 purple marbles?
Also, what is the probability you will choose 4 blue, 11 purple, and 3 black? 
for 12 purple, I did (.65)^12 x (1-.65)^6 but my answer is wrong, and similar method for second part,
thanks in advance. 
 A: Hint: what you've done is the probability of choosing 12 purple marbles first, and then 6 non-purple marbles. To fix this, you have to multiply by the number of ways you can choose which of the 18 marbles are to be purple, which is ${18 \choose 12}$.
A: You can model the problem with the multinomial distribution. Let $X=(X_i)$ and $X_i$ be the number of marbles of color $i\in$ {blue, purple, black}. Then the random vector $X$ is multinomially distributed with parameters $p_i$ and $n=18$. Thus the pmf of $X$ is $$P(X_1=x_1, X_2=x_2, X_3=x_3)=\dfrac{n!}{x_1!x_2!x_3!}0.15^{x_1}0.65^{x_2}0.2^{x_3}$$ You want to calculate the probabilities

*

*$P(X_2=12)$ and


*$P(X_1=4, X_2=11, X_3=3)$
For the first one you can avoid the summation if you consider the binomial random variable $X_2$ wich denotes the number of purple marbles only, with parameters $p=0.65$ and $n=18$. Thus the formula is $$P(X_2=12)=\dbinom{18}{12}0.65^{12}0.35^6$$
The second can be found directly by substituting in the pmf of $X$:
$$P(X_1=4, X_2=11, X_3=3)=\dfrac{18!}{4!11!3!}0.15^{4}0.65^{11}0.2^{3}$$
A: An interesting approximation / variation of part one would be to view the ratio of purple marbles to non purple marbles as $65$% vs. $35$% respectively and then "fill" a large container with that ratio so like $1000$ marbles with $650$ of them being purple and $350$ of them being nonpurple.  Then the answer would simply be :
${\binom {650} {12} * {\binom {350} 6}} / {\binom {1000} {18}}$ which is less than $1$% high vs. the correct answer. 
If you want more accuracy, increase the number of marbles so you could have:  
${\binom {6500} {12} * {\binom {3500} 6}} / {\binom {10000} {18}}$ which is less than $0.1$% high vs. the correct answer.
