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The problem you're working on is about determining which probability is needed to find the proportion of at-home games that were wins. Here's a breakdown:

### Problem Context:
- 60% of all games were home games (denoted by \( H \)).
- 25% of all games were wins (denoted by \( W \)).
- 20% of all games were both home games and wins (\( P(H \cap W) \)).

### Question:
You are asked to find the proportion of at-home games that were wins. This is essentially asking for \( P(W | H) \), which is the conditional probability that a game was a win, given that it was a home game.

### Key Points:
- To find \( P(W | H) \), the formula for conditional probability is:
  \[
  P(W | H) = \frac{P(H \cap W)}{P(H)}
  \]
- From the problem:
  - \( P(H \cap W) = 20\% \) (the proportion of all games that were both home games and wins),
  - \( P(H) = 60\% \) (the proportion of all games that were home games).

Thus, you would need \( P(H \cap W) \) and \( P(H) \) to calculate \( P(W | H) \), and the correct option would be:

- \( P(H \cap W) \) and \( P(H) \).

Let me know if you would like more details on how this is calculated or if you have any other questions!

### 5 Related Questions:
1. How is the conditional probability formula derived?
2. What are some real-life examples where conditional probability is applied?
3. Can you explain the difference between \( P(W | H) \) and \( P(H | W) \)?
4. What would happen if the problem asked for the proportion of away games that were wins?
5. How would you interpret a situation where \( P(W | H) \) equals \( P(W) \)?

### Tip:
When working with probabilities, visualizing the situation with a Venn diagram can often help clarify the relationships between different events.

Suppose a basketball team had a season of games with the following characteristics: 60% of all the games were at-home games. 25% of all games were wins. 20% of all games were at-home wins. Of the at-home games, what probabilities do you need to find to determine the proportion of at-home games that were wins?

Learn how to calculate the proportion of at-home games that were wins using conditional probability. This problem involves applying the formula P(W | H) = P(H ∩ W) / P(H) based on the given game statistics. Suitable for introductory statistics students.

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