Add interactive axioms notebook

data_science/00.02-axiomatic-probability.py

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+# /// script
+# requires-python = ">=3.10"
+# dependencies = [
+#     "marimo",
+# ]
+# ///
+
+import marimo
+
+__generated_with = "0.10.19"
+app = marimo.App()
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    title = mo.md("# 🎲 The Rules of Probability: A Beginner's Guide")
+    subtitle = mo.md("""
+    Think of axioms as the 'rules of the game' 
+    that all probabilities must follow. We'll explore these rules using simple examples.
+    """)
+
+    mo.hstack([
+        mo.vstack([title, subtitle]),
+        mo.image("https://w7.pngwing.com/pngs/774/967/png-transparent-two-white-dice-illustration-black-white-dice-bunco-dice-s-free-game-angle-black-white-thumbnail.png", width=150)
+    ])
+    return subtitle, title
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        """
+        ## The Three Fundamental Axioms
+
+        Probability theory is built on three fundamental axioms:
+
+        1. **Non-Negativity**: Probabilities can't be negative
+           - $P(A) ≥ 0$ for any event A
+
+        2. **Normalization**: The probability of the entire sample space is 1
+           - $P(S) = 1$ where S is the sample space
+
+        3. **Additivity**: For mutually exclusive events, probabilities add
+           - $P(A \cup B) = P(A) + P(B)$ when $A \cap B = \emptyset$
+
+        Let's explore these with simple examples.
+        """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        """
+        ### Understanding Axioms Through Events
+
+        Before we explore each axiom in detail, let's use a simple dice roll to see how these rules work in practice. 
+        This interactive example demonstrates all three axioms:
+
+        - All probabilities shown will be 0 or greater
+        - The probability of all possible rolls (1-6) is 1
+        - When we consider events like "rolling an even number", we're adding individual probabilities
+
+        Try different events below to see how probabilities behave according to these rules:
+        """
+    )
+    return
+
+
+@app.cell
+def _(event):
+    event
+    return
+
+
+@app.cell
+def _(mo):
+    # Single interactive example with dice
+    event = mo.ui.dropdown(
+        options=[
+            "Rolling a number from 1 to 6",
+            "Rolling a 7",
+            "Rolling an even number",
+            "Rolling a negative number"
+        ],
+        value="Rolling a number from 1 to 6",
+        label="Choose an Event"
+    )
+    return (event,)
+
+
+@app.cell
+def _(event, mo):
+    probability_map = {
+        "Rolling a number from 1 to 6": 1.0,
+        "Rolling a 7": 0.0,
+        "Rolling an even number": 0.5,
+        "Rolling a negative number": 0.0
+    }
+
+    prob = probability_map[event.value]
+
+    explanation = {
+        "Rolling a number from 1 to 6": "Certain event (must happen)",
+        "Rolling a 7": "Impossible event (can't happen)",
+        "Rolling an even number": "Possible event (might happen)",
+        "Rolling a negative number": "Impossible event (can't happen)"
+    }
+
+    mo.hstack([
+        mo.md(f"""
+        ### Event Analysis
+
+        **Probability**: {prob}
+
+        **Why?** {explanation[event.value]}
+        """),
+        mo.md(f"### Visual Scale\n{'🟦' * int(prob * 10)}{'⬜' * int((1-prob) * 10)}")
+    ])
+    return explanation, prob, probability_map
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        """
+        ## First Axiom: Non-Negativity
+
+        The first axiom states that probabilities can't be negative.
+
+        $P(A) ≥ 0$ for any event A
+
+        This makes intuitive sense - you can't have a negative chance of something happening!
+
+        **Example**: When rolling a die
+
+        - P(rolling a 6) = 1/6 ≈ 0.167 (positive)
+
+        - P(rolling a negative number) = 0 (zero, but not negative)
+        """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        """
+        ## Second Axiom: Total Probability
+
+        The second axiom states that the probability of all possible outcomes together must equal 1.
+
+        $P(S) = 1$ where S is the sample space (all possible outcomes)
+
+        **Example**: Rolling a number from 1 to 6
+
+        - P(rolling a 1) = 1/6
+        - P(rolling a 2) = 1/6
+        - P(rolling a 3) = 1/6
+        - P(rolling a 4) = 1/6
+        - P(rolling a 5) = 1/6
+        - P(rolling a 6) = 1/6
+
+        Total: 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 1
+
+        This satisfies our axiom because all possible outcomes sum to 1.
+        """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        """
+        ## Third Axiom: Additivity
+
+        For events that cannot occur simultaneously (mutually exclusive), 
+        their probabilities add.
+
+        **Example**: Rolling a die
+
+        - P(rolling a 1 or 2) = P(rolling a 1) + P(rolling a 2)
+
+        - 2/6 = 1/6 + 1/6
+
+        This works because you can't roll a 1 and 2 simultaneously.
+        """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    key_points = mo.md("""
+    ## 🎯 Key Takeaways
+
+    1. Probabilities are always between 0 and 1
+    2. Impossible events have probability 0
+    3. Certain events have probability 1
+    4. All probabilities in a sample space sum to 1
+    """)
+
+    next_topics = mo.md("""
+    ## 📚 Coming Up Next
+
+    - Addition Laws of Probability
+        - Simple Events
+        - Mutually Exclusive Events
+        - General Addition Rule
+
+    Moving towards Core Probability Laws! 🚀
+    """)
+
+    mo.hstack([
+        mo.callout(key_points, kind="info"),
+        mo.callout(next_topics, kind="success")
+    ])
+    return key_points, next_topics
+
+
+@app.cell
+def _():
+    import marimo as mo
+    return (mo,)
+
+
+if __name__ == "__main__":
+    app.run()