Problem: You’re a basketball coach, preparing your defense for the big game against the Spiders. On each play, you can either set your defense to guard the layup (2 pts) or the three pointer (3 pts). Historically, the Spiders try for layups 60% of the time, and three pointers 40% of the time. If they try the layup, it goes in 30% of the time if you defend it, and 90% of the time if you don’t. Likewise, if they try the three pointer, it goes in 25% of the time if you defend it, and 50% of the time if you don’t.
1.If you’re trying to minimize the amount of points the Spiders score per play, how often should you set up to defend the three\-pointer?
2. if the Spider coach can change the ratio of layups attempts to 3 pointer attempts, what should you do to counter that?
Here is an interact to play with the parameters$t3$, $x$, $b$, $c$, $d$, $e$, where
(0) $t3$ is the fraction of time spent shooting 3 point shots,
(1) $x$ is the fraction of time spent defending layups,
(2) $b$ is the probability of a defended layup going in,
(3) $c$ is the probability of an undefended layup going in,
(4) $d$ is the probability of a defended 3-pointer going in, and
(5) $e$ is the probability of an undefended 3-pointer going in.
By changing their values, see how the expected score per shot $ES=2\,(1-t3)\,(b\,x+c\,(1-x))+3\,t3\,(d\,(1-x)+e\,x)$ changes. In particular, is there a value for x the defenders can choose so that the Spiders cannot improve their score by changing $t3$, the fraction of time spent shooting 3s? Hint: Solve $\displaystyle\frac{\partial ES}{\partial t3}=0$. What about a value for t3 the Spiders can choose so that the defenders cannot decrease the expected score by changing $x$? Hint: Solve a partial differential equation. Bigger hint: Check the calculations in the Evaluate box above where the correct values for x and t3 are found in terms of b,c,d,e.