Trailing by two: should we go for two or three?
Documents edited by AI
Artist maxranks
Documents edited by AI
Artist maxranks
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?
- 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).
This is an AI-enhanced solution that took as input the original solution.
Let's analyze these strategies using probability theory.
- If you go for a two-pointer, the chance of winning becomes:
\[ \begin{align} \Pr \text{(win | 2-pointer)} = \Pr \text{(Overtime | 2-pointer)} \times \Pr\text{(win in overtime)} = 0.52 \times 0.5 = 0.26 \end{align} \]
Since 0.26 is less than the 0.36 chance of hitting a three, opt for the three-pointer.
- To find when you'd be indifferent:
\[ \begin{align} \Pr \text{(win | 2-pointer)} & \geq 0.36 \Leftrightarrow \\ \Pr \text{(Overtime | 2-pointer)} & \geq \frac{0.36}{0.5} = 0.72 \end{align} \]
For a deep dive into sports analytics and strategy-making, refer to Winston, Nestler, and Pelechrinis' book (2022).
Independence, Decision making
- Monty Hall Medium (Bayes rule, Conditional independence, Prior evidence)
- Red and blue balls Easy (Counting, Combinations, Repetition, Binomial)
- Repeated rolls until 4 Easy (Geometric distribution)
- Paths to destination Easy (Counting, Combinations)
- Two fair die rolls Easy (Independence, CDF, PMF)
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