  The classical ski-rental problem admits a textbook 2-competitive
deterministic algorithm, and a simple randomized algorithm that is
$\frac{e}{e-1}$-competitive in expectation. The randomized algorithm, while
optimal in expectation, has a large variance in its performance: it has more
than a 37% chance of competitive ratio exceeding 2, and a $\Theta(1/n)$ chance
of the competitive ratio exceeding $n$!
  We ask what happens to the optimal solution if we insist that the tail risk,
i.e., the chance of the competitive ratio exceeding a specific value, is
bounded by some constant $\delta$. We find that this additional modification
significantly changes the structure of the optimal solution. The probability of
purchasing skis on a given day becomes non-monotone, discontinuous, and
arbitrarily large (for sufficiently small tail risk $\delta$ and large purchase
cost $n$).
