Example 2 Choose five different numbers from $1,2, \cdots, 20$, the probability that at least two of them are consecutive numbers is $\qquad$ [1]
(2013, National High School Mathematics Joint Competition)
【Analysis】Suppose the five numbers taken out satisfy
$$
a_{1}<a_{2}<a_{3}<a_{4}<a_{5} .
$$

If these five numbers are not adjacent, then
$$
1 \leqslant a_{1}<a_{2}-1<a_{3}-2<a_{4}-3<a_{5}-4 \leqslant 16 \text {. }
$$

Thus, the number of ways to choose five non-adjacent numbers from $1,2, \cdots, 20$ is the same as the number of ways to choose five different numbers from $1,2, \cdots, 16$, which is $\mathrm{C}_{16}^{5}$.
Therefore, the required probability is $1-\frac{\mathrm{C}_{16}^{5}}{\mathrm{C}_{20}^{5}}=\frac{232}{323}$.