Let's say we have a set of discrete random variables $\{ X_1, X_2, X_3, ... X_n \}$ with known probability distributions, how to calculate the probability that $x_{max} = max(x_1, x_2, x_3 ... x_n)$ is equal to the realization of a specific random variable $x_i$? In other words, I want to calculate $Pr(x_{max}=x_i)$.
Example:
Let's say we have 4 dice $\{D_1, D_2, D_2, D_4\}$ representing 4 discrete random variables, what is the probability that $D_1$ will take the highest value after rolling the 4 dice? This probability can be denoted by $Pr(d_{max}=d_1)$
My current understanding:
First I am calculating the probability that $D_1$ is greater than $D_2, D_3, D_4$, that is, calculating $Pr(D_1 > D_2), Pr(D_1 > D_3),$ and $Pr(D_1 > D_4)$. Then $Pr(d_{max}=d_1)$ can be expressed as the product of these probabilities: $Pr(d_{max}=d_1) = Pr(D_1 > D_2) \times Pr(D_1 > D_3) \times Pr(D_1 > D_3)$
How correct is that ? My concern is that the sum of resulting probabilities is diffrent than 1, that is, $\sum\limits_{i = 1}^4 Pr(d_{max}=d_i) \ne 1$