Add probability of OR notebook

probability/03_probability_of_or.py

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+# /// script
+# requires-python = ">=3.10"
+# dependencies = [
+#     "marimo",
+#     "matplotlib",
+#     "matplotlib-venn"
+# ]
+# ///
+
+import marimo
+
+__generated_with = "0.11.2"
+app = marimo.App()
+
+
+@app.cell
+def _():
+    import marimo as mo
+    return (mo,)
+
+
+@app.cell
+def _():
+    import matplotlib.pyplot as plt
+    from matplotlib_venn import venn2
+    import numpy as np
+    return np, plt, venn2
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        # Probability of Or
+
+        When calculating the probability of either one event _or_ another occurring, we need to be careful about how we combine probabilities. The method depends on whether the events can happen together[<sup>1</sup>](https://chrispiech.github.io/probabilityForComputerScientists/en/part1/prob_or/).
+
+        Let's explore how to calculate $P(E \cup F)$ or $P(E \text{ or } F)$ in different scenarios.
+        """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ## Mutually Exclusive Events
+
+        Two events $E$ and $F$ are **mutually exclusive** if they cannot occur simultaneously. 
+        In set notation, this means:
+
+        $E \cap F = \emptyset$
+
+        For example:
+
+        - Rolling an even number (2,4,6) vs rolling an odd number (1,3,5)
+        - Drawing a heart vs drawing a spade from a deck
+        - Passing vs failing a test
+
+        Here's a Python function to check if two sets of outcomes are mutually exclusive:
+        """
+    )
+    return
+
+
+@app.cell
+def _():
+    def are_mutually_exclusive(event1, event2):
+        return len(event1.intersection(event2)) == 0
+
+    # Example with dice rolls
+    even_numbers = {2, 4, 6}
+    odd_numbers = {1, 3, 5}
+    prime_numbers = {2, 3, 5, 7}
+    return are_mutually_exclusive, even_numbers, odd_numbers, prime_numbers
+
+
+@app.cell
+def _(are_mutually_exclusive, even_numbers, odd_numbers):
+    are_mutually_exclusive(even_numbers, odd_numbers)
+    return
+
+
+@app.cell
+def _(are_mutually_exclusive, even_numbers, prime_numbers):
+    are_mutually_exclusive(even_numbers, prime_numbers)
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ## Or with Mutually Exclusive Events
+
+        For mutually exclusive events, the probability of either event occurring is simply the sum of their individual probabilities:
+
+        $P(E \cup F) = P(E) + P(F)$
+
+        This extends to multiple events. For $n$ mutually exclusive events $E_1, E_2, \ldots, E_n$:
+
+        $P(E_1 \cup E_2 \cup \cdots \cup E_n) = \sum_{i=1}^n P(E_i)$
+
+        Let's implement this calculation:
+        """
+    )
+    return
+
+
+@app.cell
+def _():
+    def prob_union_mutually_exclusive(probabilities):
+        return sum(probabilities)
+
+    # Example: Rolling a die
+    # P(even) = P(2) + P(4) + P(6)
+    p_even_mutually_exclusive = prob_union_mutually_exclusive([1/6, 1/6, 1/6])
+    print(f"P(rolling an even number) = {p_even_mutually_exclusive}")
+
+    # P(prime) = P(2) + P(3) + P(5)
+    p_prime_mutually_exclusive = prob_union_mutually_exclusive([1/6, 1/6, 1/6])
+    print(f"P(rolling a prime number) = {p_prime_mutually_exclusive}")
+    return (
+        p_even_mutually_exclusive,
+        p_prime_mutually_exclusive,
+        prob_union_mutually_exclusive,
+    )
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ## Or with Non-Mutually Exclusive Events
+
+        When events can occur together, we need to use the **inclusion-exclusion principle**:
+
+        $P(E \cup F) = P(E) + P(F) - P(E \cap F)$
+
+        Why subtract $P(E \cap F)$? Because when we add $P(E)$ and $P(F)$, we count the overlap twice!
+
+        For example, consider calculating $P(\text{prime or even})$ when rolling a die:
+        - Prime numbers: {2, 3, 5}
+        - Even numbers: {2, 4, 6}
+        - The number 2 is counted twice unless we subtract its probability
+
+        Here's how to implement this calculation:
+        """
+    )
+    return
+
+
+@app.cell
+def _():
+    def prob_union_general(p_a, p_b, p_intersection):
+        """Calculate probability of union for any two events"""
+        return p_a + p_b - p_intersection
+
+    # Example: Rolling a die
+    # P(prime or even)
+    p_prime_general = 3/6    # P(prime) = P(2,3,5)
+    p_even_general = 3/6     # P(even) = P(2,4,6)
+    p_intersection = 1/6     # P(intersection) = P(2)
+
+    result = prob_union_general(p_prime_general, p_even_general, p_intersection)
+    print(f"P(prime or even) = {p_prime_general} + {p_even_general} - {p_intersection} = {result}")
+    return (
+        p_even_general,
+        p_intersection,
+        p_prime_general,
+        prob_union_general,
+        result,
+    )
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ### Extension to Three Events
+
+        For three events, the inclusion-exclusion principle becomes:
+
+        $P(E_1 \cup E_2 \cup E_3) = P(E_1) + P(E_2) + P(E_3)$
+        $- P(E_1 \cap E_2) - P(E_1 \cap E_3) - P(E_2 \cap E_3)$
+        $+ P(E_1 \cap E_2 \cap E_3)$
+
+        The pattern is:
+
+        1. Add individual probabilities
+        2. Subtract probabilities of pairs
+        3. Add probability of triple intersection
+        """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(r"""### Interactive example:""")
+    return
+
+
+@app.cell
+def _(event_type):
+    event_type
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    # Create a dropdown to select the type of events to visualize
+    event_type = mo.ui.dropdown(
+        options=[
+            "Mutually Exclusive Events (Rolling Odd vs Even)",
+            "Non-Mutually Exclusive Events (Prime vs Even)",
+            "Three Events (Less than 3, Even, Prime)"
+        ],
+        value="Mutually Exclusive Events (Rolling Odd vs Even)",
+        label="Select Event Type"
+    )
+    return (event_type,)
+
+
+@app.cell(hide_code=True)
+def _(event_type, mo, plt, venn2):
+    # Define the events and their probabilities
+    events_data = {
+        "Mutually Exclusive Events (Rolling Odd vs Even)": {
+            "sets": (round(3/6, 2), round(3/6, 2), 0),  # (odd, even, intersection)
+            "labels": ("Odd\n{1,3,5}", "Even\n{2,4,6}"),
+            "title": "Mutually Exclusive Events: Odd vs Even Numbers",
+            "explanation": r"""
+            ### Mutually Exclusive Events
+
+            $P(\text{Odd}) = \frac{3}{6} = 0.5$
+            $P(\text{Even}) = \frac{3}{6} = 0.5$
+            $P(\text{Odd} \cap \text{Even}) = 0$
+
+            $P(\text{Odd} \cup \text{Even}) = P(\text{Odd}) + P(\text{Even}) = 1$
+
+            These events are mutually exclusive because a number cannot be both odd and even.
+            """
+        },
+        "Non-Mutually Exclusive Events (Prime vs Even)": {
+            "sets": (round(2/6, 2), round(2/6, 2), round(1/6, 2)),  # (prime-only, even-only, intersection)
+            "labels": ("Prime\n{3,5}", "Even\n{4,6}"),
+            "title": "Non-Mutually Exclusive: Prime vs Even Numbers",
+            "explanation": r"""
+            ### Non-Mutually Exclusive Events
+
+            $P(\text{Prime}) = \frac{3}{6} = 0.5$ (2,3,5)
+            $P(\text{Even}) = \frac{3}{6} = 0.5$ (2,4,6)
+            $P(\text{Prime} \cap \text{Even}) = \frac{1}{6}$ (2)
+
+            $P(\text{Prime} \cup \text{Even}) = \frac{3}{6} + \frac{3}{6} - \frac{1}{6} = \frac{5}{6}$
+
+            These events overlap because 2 is both prime and even.
+            """
+        },
+        "Three Events (Less than 3, Even, Prime)": {
+            "sets": (round(1/6, 2), round(2/6, 2), round(1/6, 2)),  # (less than 3, even, intersection)
+            "labels": ("<3\n{1,2}", "Even\n{2,4,6}"),
+            "title": "Complex Example: Numbers < 3 and Even Numbers",
+            "explanation": r"""
+            ### Complex Event Interaction
+
+            $P(x < 3) = \frac{2}{6}$ (1,2)
+            $P(\text{Even}) = \frac{3}{6}$ (2,4,6)
+            $P(x < 3 \cap \text{Even}) = \frac{1}{6}$ (2)
+
+            $P(x < 3 \cup \text{Even}) = \frac{2}{6} + \frac{3}{6} - \frac{1}{6} = \frac{4}{6}$
+
+            The number 2 belongs to both sets, requiring the inclusion-exclusion principle.
+            """
+        }
+    }
+
+    # Get data for selected event type
+    data = events_data[event_type.value]
+
+    # Create visualization
+    plt.figure(figsize=(10, 5))
+    v = venn2(subsets=data["sets"], 
+              set_labels=data["labels"])
+    plt.title(data["title"])
+
+    # Display explanation alongside visualization
+    mo.hstack([
+        plt.gcf(),
+        mo.md(data["explanation"])
+    ])
+    return data, events_data, v
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ## 🤔 Test Your Understanding
+
+        Consider rolling a six-sided die. Which of these statements are true?
+
+        <details>
+        <summary>1. P(even or less than 3) = P(even) + P(less than 3)</summary>
+
+        ❌ Incorrect! These events are not mutually exclusive (2 is both even and less than 3).
+        We need to use the inclusion-exclusion principle.
+        </details>
+
+        <details>
+        <summary>2. P(even or greater than 4) = 4/6</summary>
+
+        ✅ Correct! {2,4,6} ∪ {5,6} = {2,4,5,6}, so probability is 4/6.
+        </details>
+
+        <details>
+        <summary>3. P(prime or odd) = 5/6</summary>
+
+        ✅ Correct! {2,3,5} ∪ {1,3,5} = {1,2,3,5}, so probability is 5/6.
+        </details>
+        """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        """
+        ## Summary
+
+        You've learned:
+
+        - How to identify mutually exclusive events
+        - The addition rule for mutually exclusive events
+        - The inclusion-exclusion principle for overlapping events
+        - How to extend these concepts to multiple events
+
+        In the next lesson, we'll explore **conditional probability** - how the probability 
+        of one event changes when we know another event has occurred.
+        """
+    )
+    return
+
+
+if __name__ == "__main__":
+    app.run()