Probability of $P(X_i< X_{i-1})$ for sequence of uniform distribution

Question

Let $(X_i)_{0\le i\le n}$ be a sequence of random variable that follow uniform distribution on $\{0,\ldots,9.\}$

I would like to compute $P(X_i< X_{i-1})$ which is not difficult to see that is $$\frac{1}{100}\,\text{card}((k,l); k<l)=\frac{9}{20}.$$

It's the intuition, but how can I write $P(X_i< X_{i-1})$ to prove that is $\frac{1}{100}\,\text{card}((k,l); k<l)$ with no ambiguity ?

Answer 1

You can just write it as follows: \begin{align} P(X_i < X_{i-1}) &= \sum_{k=0}^9P(X_i <X_{i-1}|X_{i-1} = k)P(X_{i-1} = k) \\ &=\sum_{k=0}^9 \frac{k}{10}\frac{1}{10} \\ &= \frac{1}{100}\sum_{k=0}^9k\\ &= \frac{45}{100}\\ &=\frac{9}{20} \end{align}