What is Sanity check?

Example 5.6 (A range check)

“What is the chance of at least one six in six rolls?” The quick answer $6 \times \tfrac16 = 1$ fails the range check at once: the event is not certain, since all six rolls can miss. The union bound overcounts the overlaps; the complement gives $1 - (5/6)^6 \approx 0.665$.

Example 5.7 (A limit check)

A candidate proposes $\sigma\sqrt{w_1^2 + w_2^2}$ for the volatility of a two-asset portfolio with weights $w_1, w_2$ and equal volatilities $\sigma$. Setting the correlation to 1 must return $\sigma(w_1 + w_2)$, since the assets are then the same; the formula does not depend on the correlation at all, so it is the answer for correlation 0 only.