# Roll a die. Are the chances of getting two same consecutive numbers the same as getting any specific random sequence?

Let's imagine a die roll. You roll the die n times.

Example: I have a $6$ sided die. Assuming the distribution of the die is perfect, the chances of getting any single number are $1/6$. The chances of getting two consecutive same numbers (by rule of product) are $1/36$. The chances of getting any two specific numbers (let's assume a random pair like $(2,3)$) are $1/36$ also.

Let's assume that pairs of type $(n,m)$ are equivalent to pairs of type $(m,n)$ - we count them as equal. So when we roll the die n times, is the chance of some pair $(m,n)$ appearing greater than the chance of a pair $(n,n)$ appearing? That is to say is the chance of getting $(2,3)$ & $(3,2)$ in a row bigger than just the chance of getting $(4,4)$ & $(4,4)$ in a row?

When we were paying attention to the sequence in which the numbers appeared, there was just a $\frac{1}{36}$ chance of getting $(4,4),$ a $\frac{1}{36}$ chance of getting $(2,3),$ and a $\frac{1}{36}$ chance of getting $(3,2).$ When we only consider events distinct based on the number of times each face of the die has occurred, there is still only a $\frac{1}{36}$ chance that face $4$ occurred twice in two rolls, and there is still a $\frac{1}{18}$ chance that $2$ and $3$ each occurred once.
Compare this to what happens if we only write down the sum of the numbers that occur, not the actual numbers themselves. The probability that the total is $5$ is $\frac 19,$ but the probability that the total is $8$ is $\frac{5}{36},$ which is greater.