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Assume you are a basketball coach, and your team is losing by 2 points. You have the ball for the last shot. The probability of winning in overtime is roughly 50%. The probability of making a three-pointer is 36%. The probability of making a two-pointer is 52%.

Should you go for a two-pointer and take the game to overtime, or should you go for a buzzer-beating three-pointer to win the game?

Assuming everything else stays the same, adjust the probability of making a two-pointer to change the decision. What should that probability be?

We will use conditional probability and independence to solve this problem and make and data-driven decision.

- The probability of winning if you choose a two-pointer will be:

\[ \begin{align} \Pr \text{(win | 2pointer)} &= \Pr \text{(Overtime | 2pointer)} * Pr\text{(win in overtime)} \\ &= 0.52 * 0.5 = 0.26 \end{align} \]

Since 0.26 is less than 0.36, the coach should choose to go for a buzzer-beating three-pointer.

- To solve the second part, we need to solve for the two-pointer probability:

\[ \begin{align} \Pr \text{(win | 2pointer)} & \geq 0.36 \Leftrightarrow \\ \Pr \text{(Overtime | 2pointer)} & \geq 0.36/Pr\text{(win in overtime)} \\ & = 0.36/0.5 \\ & = 0.72 \end{align} \]

If you are interested in sports analytics and decision-making, check out the book by Winston, Nestler, and Pelechrinis (2022).

Independence, Decision making