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The image is asking to find the **Banzhaf power distribution** for a weighted voting system, with the weights given as \([32, 17, 15, 13, 1]\). To solve this, we need to calculate the Banzhaf power index for each player by considering how many times each player is pivotal in various coalitions. 

### Steps:
1. **Determine total voting weight**: The sum of all weights is \( 32 + 17 + 15 + 13 + 1 = 78 \).
2. **Quota for majority**: The quota is more than half of the total weight, so \( \text{Quota} = \lceil \frac{78}{2} \rceil = 39 \). Thus, any coalition of voters must have a combined weight of at least 39 to win.
3. **Identify pivotal players**: For each coalition, we will check if a player's removal from the coalition changes it from a winning to a losing coalition. This is the pivotal player in that coalition.
4. **Calculate power index**: For each player, count how many times they are pivotal and divide this by the total number of pivotal instances across all players.

I'll now calculate the Banzhaf power distribution based on these principles.

Let's proceed with that calculation.The Banzhaf power distribution for the given weighted voting system \([32, 17, 15, 13, 1]\) is as follows:

- **Player \(P_1\)** (Weight: 32): Power = \( \frac{12}{24} = 0.5 \)
- **Player \(P_2\)** (Weight: 17): Power = \( \frac{4}{24} = 0.1667 \)
- **Player \(P_3\)** (Weight: 15): Power = \( \frac{4}{24} = 0.1667 \)
- **Player \(P_4\)** (Weight: 13): Power = \( \frac{4}{24} = 0.1667 \)
- **Player \(P_5\)** (Weight: 1): Power = \( \frac{0}{24} = 0 \)

### Questions for deeper exploration:
1. What happens to the power distribution if the weights of the players are adjusted?
2. How would the power change if the quota is increased to 40 instead of 39?
3. Can Player \(P_5\) gain any power if the total weight is redistributed differently?
4. How does the Banzhaf index compare to other power indices like Shapley-Shubik?
5. In what types of real-world voting systems might this method be useful?

**Tip:** The Banzhaf index is sensitive to small changes in voting weight, making it a useful tool for evaluating influence in committee decisions.

Find the Banzhaf power distribution of the weighted voting system [32, 17, 15, 13, 1]. Give each player's power as a fraction or decimal value.

Learn how to calculate the Banzhaf power distribution for the weighted voting system [32, 17, 15, 13, 1]. This step-by-step solution explains how to find each player's power index as a fraction or decimal based on their pivotal role in winning coalitions. Suitable for students in advanced high school or college-level math courses.

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