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Exercise 7:
Given a 6-sided fair dice you try a new game. You only throw the dice a second time if the result of the first throw is lower or equal to 3. Which of the following probability mass functions will be the one you should get given this new constraint?
Hints:

You can simulate the second throws as a numpy array and then make the values that met a certain criteria equal to 0 by using np.where

The exercise is about determining the probability mass function (PMF) for a modified dice-throwing game with a specific rule. Here's the rule:

1. You throw a 6-sided fair die (with faces numbered 1 through 6).
2. You throw the die a second time only if the result of the first throw is less than or equal to 3.

To solve this problem, let's analyze the possible outcomes:

- **First Throw (T1):** Since the die is fair, the probability of rolling any number from 1 to 6 is equal. That is \( P(T1 = k) = \frac{1}{6} \) for \( k \in \{1, 2, 3, 4, 5, 6\} \).

- **Second Throw (T2):** 
  - If the first throw results in 1, 2, or 3, you roll the die again. 
  - If the first throw results in 4, 5, or 6, you do not roll again.

The task is to determine which of the provided histograms represents the PMF under this new rule. Here's how to simulate this:

1. **Count the Effective Outcomes:** 
   - Outcomes 1, 2, and 3 will include results from a second throw. Each of these has an equal chance of triggering a second throw.
   - Outcomes 4, 5, and 6 are final, as no second throw occurs after these results.

2. **Simulate the Experiment:**
   - Using an array to represent first throw outcomes.
   - For results 1, 2, and 3, use a second throw (replace these with second throw outcomes).
   - For results 4, 5, and 6, keep them as is.

3. **Calculate the PMF:**
   - The final PMF is a combination of both first and (if needed) second throws.

### Steps for Python Simulation with `numpy` (using the hint provided):

```python
import numpy as np

# Simulate 10,000 trials for the first throw
first_throw = np.random.randint(1, 7, size=10000)

# Simulate the second throw for cases where the first throw is less than or equal to 3
second_throw = np.random.randint(1, 7, size=10000)

# Only keep second throws for first throws of 1, 2, or 3
final_results = np.where(first_throw <= 3, second_throw, first_throw)

# Count occurrences for each possible outcome (1 through 6)
values, counts = np.unique(final_results, return_counts=True)

# Normalize to get the probability mass function (PMF)
pmf = counts / sum(counts)
```

### Explanation of Expected PMF:

- Numbers 4, 5, and 6 will each have a probability of roughly \(\frac{1}{6}\).
- Numbers 1, 2, and 3 might appear more frequently since a second throw can lead to these numbers appearing again.

### Matching Histogram to Results:

- To match the histograms, look for:
  - Higher probabilities in the range 1 to 3 than in 4 to 6.
  - Equal or nearly equal probabilities for 4, 5, and 6.

The histogram that matches these characteristics would correctly represent the PMF of the dice game under the given constraints.

Would you like more details about the simulation, or do you have specific questions about this problem?

Here are some related questions for deeper understanding:

1. How would the PMF change if the condition for the second throw was altered to be "less than 4" instead of "less than or equal to 3"?
2. What is the expected value of a single game under this new rule?
3. How would you calculate the variance of this distribution?
4. How does this modified dice game compare to rolling two dice independently?
5. Can you derive a general formula for the PMF for any condition placed on the second throw?

**Tip:** Simulating random experiments and comparing the results to theoretical predictions is a powerful method for understanding probability and statistics.

Explore how to analyze probability mass functions (PMF) for a modified dice game using conditional probabilities and simulations. Learn about the impact of specific conditions on PMFs and how to simulate outcomes effectively. Suitable for advanced high school students and learners interested in probability theory.

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