How to calculate probability using multinomial distribution?

So according to the multinomial distribution, the probability function $\Pr(X_1 = x_1, X_2 = x_2, \dots, X_k = x_k)$ is equal to $\dfrac{n!}{x_1! x_2! \cdots x_k!} \cdot p_1^{x_1}\cdot p_2^{x_2} \cdots p_k^{x_k}$

And if we were to sum all of the probabilites for all possible values of $x_1, x_2, \dots x_k$, we would get one.

The thing is that summing all of those probablities also includes the situations in which at least one $x_j = 0$. How would I go about computing the following probablity function?:

$$\Pr(X_1=x_1>0, X_2=x_2>0, \ldots, X_k=x_k>0) = ?$$

So basically, the sum of all probabilities of the scenarios in which no $x_j = 0$ and the sum of all the probabilities of the scenarios in which at least one $x_j = 0$ is equal to $1$.

I want to find the sum of all probabilities of the scenarios in which no $x_j = 0$

• Subtract one from everything and use the multinomial distribution with $n-k$ instead of $n$. – Zach Jul 11 '13 at 18:25
• Do you mean subtract one from all of the xj's? – Robert Jefferies Jul 11 '13 at 18:26
• You say "The problem is . . .", but that doesn't seem to be a problem at all. If you include all the possible values of $x_1,\ldots,x_n$, all nonnegative integers, satisfying $x_1+\cdots+x_k=n$, including the cases where some of the $x_j$ are $0$, you get $1$. – Michael Hardy Jul 11 '13 at 18:40
• @Zach could you please elaborate a little but more? – Robert Jefferies Jul 11 '13 at 19:04
• @Zach could you please elaborate a little more? – Robert Jefferies Jul 12 '13 at 13:07

You say "The problem is . . .", but that doesn't seem to be a problem at all. If you include all the possible values of $x_1,\ldots,x_n$, all nonnegative integers, satisfying $x_1+\cdots+x_k=n$, including the cases where some of the $x_j$ are $0$, you get $1$.
• I understand, but I dont want to include the possible values of $x_1, x_2, ... , x_k$ for nonnegative integers, I only want positive integers. The value I should get should be less than 1. – Robert Jefferies Jul 11 '13 at 18:47
• The real problem is trying to compute the sum of the probabilities for which no $x_j = 0$ – Robert Jefferies Jul 11 '13 at 18:57