An index card simulation for the penalty kick problem you described can help model how goalkeepers decide where to dive (left, right, or center) and how successful they are in stopping the shot. The simulation can use randomness to simulate penalty outcomes and decisions, allowing students to observe patterns and draw conclusions based on probabilities.

Steps to run the index card simulation:

1. Set Up the Simulation Variables:

• Player Decisions: There are three choices for the goalkeeper: dive left, dive right, or stay in the center.
• Shooter Behavior: Penalty shooters can aim their shot either to the left, right, or center.
• Success Conditions: The goalkeeper will stop the ball only if they dive in the correct direction (i.e., matching the shot direction).

2. Create the Index Cards:

• Goalkeeper’s Actions: Prepare a deck of index cards for the goalkeeper’s actions, where each card represents a possible action:
  • ⅓ of the cards say “Dive Left”
  • ⅓ of the cards say “Dive Right”
  • ⅓ of the cards say “Stay Center”
• Shooter’s Actions: Prepare a second deck of index cards for the shooter’s actions:
  • ⅓ of the cards say “Shoot Left”
  • ⅓ of the cards say “Shoot Right”
  • ⅓ of the cards say “Shoot Center”

3. Simulate the Game:

• Shuffle both decks (goalkeeper and shooter) separately.
• Draw one card from the goalkeeper deck and one card from the shooter deck for each simulated penalty kick.
• Outcome Determination: Compare the cards:
  • If the goalkeeper’s choice matches the shooter’s choice, the shot is saved.
  • If they don’t match, the shot results in a goal.

4. Record the Results:

• After each simulated penalty kick, record whether the goalkeeper made the save or not.
• Repeat this process for a sufficient number of trials (e.g., 50–100 penalty kicks) to collect data.

5. Analyze the Results:

• Success Rate: Calculate the percentage of penalties saved by the goalkeeper.
• Explore Variations: You could introduce different shooter behaviors, such as more frequent shots to a particular side, and adjust the probabilities for certain shooter decisions.

Additional Variations:

• Goalkeeper Bias: Some goalkeepers might dive left or right more often. Adjust the goalkeeper deck to reflect such biases.
• Shooter Bias: Similarly, some shooters might favor one side over the other.

Tip for Discussion:

Ask students to reflect on how real-life goalkeepers might adjust their strategy if they know the shooter’s tendencies (for example, a shooter who tends to favor the right side).

---

Would you like to dive into any specific aspect of the simulation, like how to analyze or expand it? Here are five related questions to consider:

1. How would changing the probabilities of diving left, right, or staying center affect the simulation outcomes?
2. What if the shooter has a known preference for shooting left? How would you modify the index card deck for that?
3. How many trials (penalty kicks) would you need to simulate for the data to be statistically significant?
4. How would you introduce different player skill levels in the simulation?
5. Could this simulation be modified for use in a different sport or situation? How?

Tip:

Always run enough trials to reduce random chance effects, ensuring that your simulation data is a reliable reflection of the real-world scenario you're modeling.