The book of Santosh Venkatesh The Theory of Probability, Explorations and Applications, Cambridge University Press, 2012 has many interesting examples that would be accessible to a good high school student. Let me mention a few to give you a taste.
A beautiful probabilistic proof of Cayley's formula on the number of trees.
A nice connection between Viete's formula
$$ \frac{\sin x}{x} =\prod_{k=1}^\infty \cos\left(\frac{x}{2^k}\right) $$
and the distribution of digits in the binary expansion of a random number $x\in [0,1]$.
Random graphs.
Limits on wireless broadcasting.
The problems at the end of each chapter are excellent, of varied degrees of difficulty.