To Foul or Not to Foul?

Assume that an average team scores exactly $1$ point per offensive possession, a figure that accounts for multiple shots if the team rebounds its own miss (or misses) on a single trip, and that it rebounds $15$ percent of its own missed free throws.

Now suppose you are the coach of a team playing an average opponent that’s in the bonus. The other team has the ball, the game is tight, and you want to minimize the expected number of points your opponent will earn on this particular possession. How low does the ball-handler’s free throw shooting percentage need to be for you to instruct your team to foul that player (when they are not in the act of shooting)?

Let's let $r$ be the average team's rebounding percentage on missed free throws that are in play, and let $p$ be the ball handler's free throw percentage. In the case where you don't foul, the expected points for this possession is $1.$ So all we need to do is set up the expected points if you do foul the ball handler on the floor and figure out for what values of $p$ that expectation is less than $1$.

For the time being I will assume that you have instructed the team that if a free throw is missed then do not foul, so that we can count the expected number of points in the event that the opposing team rebounds a missed free throw as the usual expected points per offensive possession $1.$ If there is a foul, then there are five outcomes:

(i) both free throws are made, which results in $2$ additional points and has probability $p^2$;

(ii) the first free throw is made, the second one is missed and the opposing team rebounds, which results in an expected $2$ additional points and has probability $p(1-p)r$;

(iii) the first free throw is made, the second one is missed and your team rebounds, which results in $1$ additional point and has probability $p(1-p)(1-r)$;

(iv) the first free throw is missed and the opposing team rebounds, which results in an expected $1$ additional point and has probability $(1-p)r$;

and (v) the first free throw is missed and your team rebounds, which results in $0$ points and has probability $(1-p)(1-r).$

Taking the expectation, we get \begin{align*} E(p,r) &= 2p^2 + 2p(1-p)r + p(1-p)(1-r) + (1-p)r + 0(1-p)(1-r)\\ &= (2-2r-(1-r))p^2 + (2r+(1-r) -r)p + r\\ &= (1-r)p^2 + p + r. \end{align*} So, solving the quadratic, we have that it is better to foul the ball handler (assuming no subsequent fouls if the opposing team rebounds) whenever $E(p,r) \leq 1,$ or $$0 \leq p \leq \frac{-1 + \sqrt{1 + 4(1-r)^2}}{2(1-r)}.$$ When $r = 0.15$ for an average team, you should instruct your team to foul the ball handlers with free throw percentages less than or equal to $\frac{-1 + \sqrt{1 + 4 \cdot 0.85^2}}{2 \cdot 0.85} = 0.5719.$