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Question for you, describe how you would run an index card simulation for this problem. This is a sport statistic math class

To run an **index card simulation** for this penalty kick scenario in a sports statistics class, you can model the situation using index cards that represent the outcomes of penalty kicks based on the data in the image. Here's how to go about it step by step:

### 1. Define Categories and Outcomes
You have two main actions the goalkeeper can take based on the table:
- **Stay** (goalkeeper stays in the center)
- **Dive** (goalkeeper dives either left or right)

For each action, there are two possible outcomes:
- **Stopped** (the goalkeeper saves the penalty)
- **Scored** (the goalkeeper fails to save, and the ball goes into the net)

### 2. Prepare the Index Cards
Use index cards to represent each outcome. Based on the data, you can assign the appropriate number of cards for each action.

#### Total Number of Cards
From the table, you have:
- **18 total kicks** where the keeper stays.
- **268 total kicks** where the keeper dives.

This means you'll need 18 cards for when the keeper stays, and 268 cards for when the keeper dives.

#### Outcome Proportions
For each of these groups (Stay and Dive), divide the cards based on the "Stopped" and "Scored" outcomes.

1. **Stay (18 total kicks):**
   - **6 stopped**: Create 6 cards labeled "Stay - Stopped."
   - **12 scored**: Create 12 cards labeled "Stay - Scored."

2. **Dive (268 total kicks):**
   - **36 stopped**: Create 36 cards labeled "Dive - Stopped."
   - **232 scored**: Create 232 cards labeled "Dive - Scored."

### 3. Shuffle and Simulate
Once you've prepared the index cards with these proportions, you can conduct a random simulation:

1. **Shuffle** all the cards together, representing a randomized selection of what happens in a real-world penalty kick scenario.
2. **Draw a card** to simulate the outcome of a penalty kick:
   - If you draw a "Stay" card, it represents the keeper staying in the center. Depending on whether the card says "Stopped" or "Scored," that’s the result of the simulation.
   - If you draw a "Dive" card, it represents the keeper diving, and you note whether the outcome is "Stopped" or "Scored."

### 4. Repeat for Multiple Trials
To get a meaningful simulation, repeat this process for many trials (at least 50 to 100 draws). Afterward, you can calculate the **stopping percentage** and compare it to the original data.

### 5. Analyze Results
Once you've run the simulation several times, record the number of successful saves and goals allowed. You can calculate the percentages for each action and compare them to the actual data.

For instance:
- What percentage of kicks were saved when the keeper stayed versus dived?
- How closely do the simulated results match the real data (66.67% scored when staying, 86.57% scored when diving)?

### Tip for Running a Simulation
- Make sure you use enough trials to smooth out random fluctuations. The larger the number of trials, the closer your simulation will get to the actual percentages.

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**Would you like more details on running simulations or analyzing the outcomes?**

### Relative questions:
1. How would you compute the probability of scoring using the simulation results?
2. How could you modify the simulation to reflect different strategies by the goalkeeper?
3. What other factors could be included to make the simulation more realistic?
4. How would you present the results of your simulation in a report or a presentation?
5. What assumptions are made in this index card simulation, and how might they affect the accuracy?

**Tip:** When running simulations, ensure you have a large enough sample size to minimize random variations and obtain accurate results.

Explore how to run an index card simulation to model goalkeeper behavior during penalty kicks using probability and sports statistics. This lesson includes a step-by-step approach to creating simulations, analyzing outcomes, and comparing them with real-world data. Ideal for Grades 10-12.

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