A set of $n$ people are given some sweets. There is candy A, B and C and each is given with probability $p_A,p_B$ and $p_C$.
I am trying to find the possible combinations of this system.
We can use the multinomial theorem, assuming that a fraction $i_X$ receives candy $X\in A,B,C$ such that:
$$
\sum_{i_{X}=n}
\begin{pmatrix}
n\\
i_A,i_B,i_{C}, i
\end{pmatrix}
p_A^{i_A}
p_B^{i_B}
p_C^{i_{C}}
[1-p_A-p_B-p_{C})]^{i}
$$
where the sum is over all the combinations of $i_X$ that sum to $n$. This leads to:
$$
(p_A + p_B + p_C + (1-p_A-p_B-p_{C}))^n
$$
However, I don't see why the following is also not a valid approach to this question.
$$
\sum^{n}_{i_A=0}
\begin{pmatrix}
n \\ i_A
\end{pmatrix}
p_A^{i_A} (1-p_A)^{n-i_{A}}
\sum^{n-i_A}_{i_{B}=0}
\begin{pmatrix}
n-i_A \\
i_B
\end{pmatrix}
p_B^{i_{B}}(1-p_B)^{n-i_{A}-i_{B}}
\sum^{n-i_{A}-i_{B}}_{i_{C}=0}
\begin{pmatrix}
n-i_{A}-i_{B}\\
i_{C}
\end{pmatrix}
p_{C}^{i_{C}}(1-p_{C})^{n-i_{A}-i_{B}-i_{C}}
$$
which leads to
$$
[p_A + (1-p_A)\big[ p_B + (1-p_B)[p_{C} +(1-p_{C})]\big]]^{n}
$$
Which is essentially the binomial theorem used in succession for the success or fail of each candy.
What is the difference between each approach?
Best Answer
Let $X_A,X_B,X_C$ denote the number of persons that receive a candy of sort $A,B,C$ respectively. Then $$P(X_A=i_A,X_B=i_B,X_C=i_C)=\binom{n}{i_A,i_B,i_C,i}p_A^{i_A}p_B^{i_B}p_C^{i_C}(1-p_A-p_B-p_C)^i$$ where $i_A+i_B+i_C+i=n$.
We can also write $$P(X_A=i_A,X_B=i_B,X_C=i_C)=$$$$P(X_A=i_A)P(X_B=i_B\mid X_A=i_A)P(X_C=i_C\mid X_A=i_A, X_B=i_B)=$$$$\binom{n}{i_A}p_A^{i_A}(1-p_A)^{n-i_A}\times$$$$\binom{n-i_A}{i_B}\left(\frac{p_B}{1-p_A}\right)^{i_B}\left(1-\frac{p_B}{1-p_A}\right)^{n-i_A-i_B}\times$$$$\binom{n-i_A-i_B}{i_C}\left(\frac{p_C}{1-p_A-p_B}\right)^{i_C}\left(1-\frac{p_C}{1-p_A-p_B}\right)^{i}$$ where again $i_A+i_B+i_C+i=n$.
If under e.g. condition that $i_A$ persons receive candy of sort $A$ and we are focusing on the question how many under this condition will receive candy of sort $B$ then we are dealing with a binomial distribution with parameters $n-i_A$ and $\frac{p_B}{1-p_A}$ (not $p_B$ as you seem to suggest).