Add probability axioms notebook

probability/02_axioms.py

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+# /// script
+# requires-python = ">=3.11"
+# dependencies = [
+#     "marimo",
+#     "matplotlib==3.10.0",
+#     "numpy==2.2.2",
+# ]
+# ///
+
+import marimo
+
+__generated_with = "0.11.2"
+app = marimo.App(width="medium")
+
+
+@app.cell
+def _():
+    import marimo as mo
+    return (mo,)
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        # Axioms of Probability
+
+        Probability theory is built on three fundamental axioms, known as the [Kolmogorov axioms](https://en.wikipedia.org/wiki/Probability_axioms). These axioms form 
+        the mathematical foundation for all of probability theory[<sup>1</sup>](https://chrispiech.github.io/probabilityForComputerScientists/en/part1/probability).
+
+        Let's explore each axiom and understand why they make intuitive sense:
+        """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ## The Three Axioms
+
+        | Axiom | Mathematical Form | Meaning |
+        |-------|------------------|----------|
+        | **Axiom 1** | $0 \leq P(E) \leq 1$ | All probabilities are between 0 and 1 |
+        | **Axiom 2** | $P(S) = 1$ | The probability of the sample space is 1 |
+        | **Axiom 3** | $P(E \cup F) = P(E) + P(F)$ | For mutually exclusive events, probabilities add |
+
+        where the set $S$ is the sample space (all possible outcomes), and $E$ and $F$ are sets that represent events. The notation $P(E)$ denotes the probability of $E$, which you can interpret as the chance that something happens. $P(E) = 0$ means that the event cannot happen, while $P(E) = 1$ means the event will happen no matter what; $P(E) = 0.5$ means that $E$ has a 50% chance of happening.
+
+        For an example, when rolling a fair six-sided die once, the sample space $S$ is the set of die faces ${1, 2, 3, 4, 5, 6}$, and there are many possible events; we'll see some examples below.
+        """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ## Understanding Through Examples
+
+        Let's explore these axioms using a simple experiment: rolling a fair six-sided die.
+        We'll use this to demonstrate why each axiom makes intuitive sense.
+        """
+    )
+    return
+
+
+@app.cell
+def _(event):
+    event
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    # Create an interactive widget to explore different events
+
+    event = mo.ui.dropdown(
+
+        options=[
+
+            "Rolling an even number (2,4,6)",
+
+            "Rolling an odd number (1,3,5)",
+
+            "Rolling a prime number (2,3,5)",
+
+            "Rolling less than 4 (1,2,3)",
+
+            "Any possible roll (1,2,3,4,5,6)",
+
+        ],
+
+        value="Rolling an even number (2,4,6)",
+
+        label="Select an event"
+
+    )
+    return (event,)
+
+
+@app.cell(hide_code=True)
+def _(event, mo, np, plt):
+    # Define the probabilities for each event
+    event_map = {
+        "Rolling an even number (2,4,6)": [2, 4, 6],
+        "Rolling an odd number (1,3,5)": [1, 3, 5],
+        "Rolling a prime number (2,3,5)": [2, 3, 5],
+        "Rolling less than 4 (1,2,3)": [1, 2, 3],
+        "Any possible roll (1,2,3,4,5,6)": [1, 2, 3, 4, 5, 6],
+    }
+
+    # Get outcomes directly from the event value
+    outcomes = event_map[event.value]
+    prob = len(outcomes) / 6
+
+    # Visualize the probability
+    dice = np.arange(1, 7)
+    colors = ['#1f77b4' if d in outcomes else '#d3d3d3' for d in dice]
+
+    fig, ax = plt.subplots(figsize=(8, 2))
+    ax.bar(dice, np.ones_like(dice), color=colors)
+    ax.set_xticks(dice)
+    ax.set_yticks([])
+    ax.set_title(f"P(Event) = {prob:.2f}")
+
+    # Add explanation
+    explanation = mo.md(f"""
+    **Event**: {event.value}
+
+    **Probability**: {prob:.2f}
+
+    **Favorable outcomes**: {outcomes}
+
+    This example demonstrates:
+
+    - Axiom 1: The probability is between 0 and 1
+
+    - Axiom 2: For the sample space, P(S) = 1
+
+    - Axiom 3: The probability is the sum of individual outcome probabilities
+    """)
+
+    mo.hstack([plt.gcf(), explanation])
+    return ax, colors, dice, event_map, explanation, fig, outcomes, prob
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ## Why These Axioms Matter
+
+        These axioms are more than just rules - they provide the foundation for all of probability theory:
+
+        1. **Non-negativity** (Axiom 1) makes intuitive sense: you can't have a negative number of occurrences 
+           in any experiment.
+
+        2. **Normalization** (Axiom 2) ensures that something must happen - the total probability must be 1.
+
+        3. **Additivity** (Axiom 3) lets us build complex probabilities from simple ones, but only for events 
+           that can't happen together (mutually exclusive events).
+
+        From these simple rules, we can derive all the powerful tools of probability theory that are used in 
+        statistics, machine learning, and other fields.
+        """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ## 🤔 Test Your Understanding
+
+        Consider rolling two dice. Which of these statements follow from the axioms?
+
+        <details>
+        <summary>1. P(sum is 13) = 0</summary>
+
+        ✅ Correct! This follows from Axiom 1. Since no combination of dice can sum to 13, 
+        the probability must be non-negative but can be 0.
+        </details>
+
+        <details>
+        <summary>2. P(sum is 7) + P(sum is not 7) = 1</summary>
+
+        ✅ Correct! This follows from Axioms 2 and 3. These events are mutually exclusive and cover 
+        the entire sample space.
+        </details>
+
+        <details>
+        <summary>3. P(first die is 6 or second die is 6) = P(first die is 6) + P(second die is 6)</summary>
+
+        ❌ Incorrect! This doesn't follow from Axiom 3 because the events are not mutually exclusive - 
+        you could roll (6,6).
+        </details>
+        """
+    )
+    return
+
+
+@app.cell
+def _():
+    import numpy as np
+    import matplotlib.pyplot as plt
+    return np, plt
+
+
+if __name__ == "__main__":
+    app.run()