# Probability of the highest roll in two die pools being the same

Say we roll n identical, fair dice, each with d sides. (Every side comes up with the same probability.) On each die, the sides are numbered from $1$ to d with no repeating numbers, as you would expect. So it's an ordinary d sided die pool.

How would we calculate the odds of the highest rolled die value from a given dice pool equaling the highest rolled die value from a different dice pool?

• You can use the order-statistic formulas in this wikipedia article to compute the probability of a specific number being the max roll. I’m not sure that there’s any more convenient way to work out the probabilities of matching values of two rolls besides working through the cases. – amd Oct 3 '17 at 2:54

Let $(X_i)_{i=1}^n, (Y_i)_{i=1}^n$ be the sequence of results from the individual die, which are mutually-independent and identically uniformly-discrete-distributed random values.
\begin{align}&\qquad\mathsf P(\max{(X_i)}_{i=1}^d=\max{(Y_i)}_{i=1}^d) \\[1ex]&=~ \sum_{k=1}^d \mathsf P(\max{(X_i)}_{i=1}^d=k)\cdot\mathsf P(\max{(Y_i)}_{i=1}^d=k)\\[1ex] &=~\sum_{k=1}^d \mathsf P\left(\bigcap_{i=1}^d \{X_i\leq k\} \smallsetminus\bigcap_{j=1}^d \{X_j\leq (k-1)\}\right)\cdotp\mathsf P\left(\bigcap_{i=1}^d \{Y_i\leq k\} \smallsetminus\bigcap_{j=1}^d \{Y_j\leq (k-1)\})\right)\\[1ex] &=~\sum_{k=1}^d \left(\mathsf P(X_1\leq k)^d -\mathsf P(X_1\leq (k-1))^d\right)\cdotp\left(\mathsf P(Y_1\leq k)^d -\mathsf P(Y_1\leq (k-1))^d\right)\\[1ex] &=~\sum_{k=1}^d \dfrac{(k^n-(k-1)^n)^2}{d^{2n}}\\[1ex]&~\ddots\end{align}