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.

1. A beautiful probabilistic proof of Cayley's formula on the number of trees.

2. 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 $x\in [0,1]$.

3. Random graphs.

4. Limits on wireless broadcasting.

The problems at the end of each chapter are excellent, of varied degrees of difficulty.