Add conditional probability notebook

Author: Haleshot

probability/04_conditional_probability.py

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+# /// script
+# requires-python = ">=3.10"
+# dependencies = [
+#     "marimo",
+#     "matplotlib==3.10.0",
+#     "matplotlib-venn==1.1.1",
+#     "numpy==2.2.2",
+# ]
+# ///
+
+import marimo
+
+__generated_with = "0.11.2"
+app = marimo.App(width="medium", app_title="Conditional Probability")
+
+
+@app.cell
+def _():
+    import marimo as mo
+    return (mo,)
+
+
+@app.cell
+def _():
+    import matplotlib.pyplot as plt
+    from matplotlib_venn import venn3
+    import numpy as np
+    return np, plt, venn3
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        # Conditional Probability
+
+        In probability theory, we often want to update our beliefs when we receive new information. 
+        Conditional probability helps us formalize this process by calculating "_what is the chance of 
+        event $E$ happening given that we have already observed some other event $F$?_"[<sup>1</sup>](https://chrispiech.github.io/probabilityForComputerScientists/en/part1/cond_prob/)
+
+        When we condition on an event $F$:
+
+        - We enter the universe where $F$ has occurred
+        - Only outcomes consistent with $F$ are possible
+        - Our sample space reduces to $F$
+        """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ## Definition of Conditional Probability
+
+        The probability of event $E$ given that event $F$ has occurred is denoted as $P(E|F)$ and is defined as:
+
+        $$P(E|F) = \frac{P(E \cap F)}{P(F)}$$
+
+        This formula tells us that the conditional probability is the probability of both events occurring 
+        divided by the probability of the conditioning event.
+
+        Let's understand this with a function that computes conditional probability:
+        """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo, plt, venn3):
+    # Create figure with square boundaries
+    plt.figure(figsize=(10, 5))
+
+    # Draw square sample space first
+    rect = plt.Rectangle((-2, -2), 4, 4, fill=False, color='gray', linestyle='--')
+    plt.gca().add_patch(rect)
+
+    # Set the axis limits to show the full rectangle
+    plt.xlim(-2.5, 2.5)
+    plt.ylim(-2.5, 2.5)
+
+    # Create Venn diagram showing E and F
+    # For venn3, subsets order is: (100, 010, 110, 001, 101, 011, 111)
+    # Representing: (A, B, AB, C, AC, BC, ABC)
+    v = venn3(subsets=(30, 20, 10, 40, 0, 0, 0),
+             set_labels=('E', 'F', 'Rest'))
+
+    # Customize colors
+    if v:
+        for id in ['100', '010', '110', '001']:
+            if v.get_patch_by_id(id):
+                if id == '100':
+                    v.get_patch_by_id(id).set_color('#ffcccc')  # Light red for E
+                elif id == '010':
+                    v.get_patch_by_id(id).set_color('#ccffcc')  # Light green for F
+                elif id == '110':
+                    v.get_patch_by_id(id).set_color('#e6ffe6')  # Lighter green for intersection
+                elif id == '001':
+                    v.get_patch_by_id(id).set_color('white')    # White for rest
+
+    plt.title('Conditional Probability in Sample Space')
+
+    # Remove ticks but keep the box visible
+    plt.gca().set_yticks([])
+    plt.gca().set_xticks([])
+    plt.axis('on')
+
+    # Add sample space annotation with arrow
+    plt.annotate('Sample Space (100)', 
+                xy=(-1.5, 1.5),
+                xytext=(-2.2, 2),
+                bbox=dict(boxstyle='round,pad=0.5', fc='white', ec='gray'),
+                arrowprops=dict(arrowstyle='->'))
+
+    # Add explanation
+    explanation = mo.md(r"""
+    ### Visual Intuition
+
+    In our sample space of 100 outcomes:
+
+    - Event $E$ occurs in 40 cases (red region: 30 + 10)
+    - Event $F$ occurs in 30 cases (green region: 20 + 10)
+    - Both events occur together in 10 cases (overlap)
+    - Remaining cases: 40 (to complete sample space of 100)
+
+    When we condition on $F$:
+    $$P(E|F) = \frac{P(E \cap F)}{P(F)} = \frac{10}{30} = \frac{1}{3} \approx 0.33$$
+
+    This means: When we know $F$ has occurred (restricting ourselves to the green region),
+    the probability of $E$ also occurring is $\frac{1}{3}$ - as 10 out of the 30 cases in the 
+    green region also belong to the red region.
+    """)
+
+    mo.hstack([plt.gcf(), explanation])
+    return explanation, id, rect, v
+
+
+@app.cell
+def _():
+    def conditional_probability(p_intersection, p_condition):
+        if p_condition == 0:
+            raise ValueError("Cannot condition on an impossible event")
+        if p_intersection > p_condition:
+            raise ValueError("P(E∩F) cannot be greater than P(F)")
+
+        return p_intersection / p_condition
+    return (conditional_probability,)
+
+
+@app.cell
+def _(conditional_probability):
+    # Example 1: Rolling a die
+    # E: Rolling an even number (2,4,6)
+    # F: Rolling a number greater than 3 (4,5,6)
+    p_even_given_greater_than_3 = conditional_probability(2/6, 3/6)
+    print("Example 1: Rolling a die")
+    print(f"P(Even | >3) = {p_even_given_greater_than_3}")  # Should be 2/3
+    return (p_even_given_greater_than_3,)
+
+
+@app.cell
+def _(conditional_probability):
+    # Example 2: Cards
+    # E: Drawing a Heart
+    # F: Drawing a Face card (J,Q,K)
+    p_heart_given_face = conditional_probability(3/52, 12/52)
+    print("\nExample 2: Drawing cards")
+    print(f"P(Heart | Face card) = {p_heart_given_face}")  # Should be 1/4
+    return (p_heart_given_face,)
+
+
+@app.cell
+def _(conditional_probability):
+    # Example 3: Student grades
+    # E: Getting an A
+    # F: Studying more than 3 hours
+    p_a_given_study = conditional_probability(0.24, 0.40)
+    print("\nExample 3: Student grades")
+    print(f"P(A | Studied >3hrs) = {p_a_given_study}")  # Should be 0.6
+    return (p_a_given_study,)
+
+
+@app.cell
+def _(conditional_probability):
+    # Example 4: Weather
+    # E: Raining
+    # F: Cloudy
+    p_rain_given_cloudy = conditional_probability(0.15, 0.30)
+    print("\nExample 4: Weather")
+    print(f"P(Rain | Cloudy) = {p_rain_given_cloudy}")  # Should be 0.5
+    return (p_rain_given_cloudy,)
+
+
+@app.cell
+def _(conditional_probability):
+    # Example 5: Error cases
+    print("\nExample 5: Error cases")
+    try:
+        # Cannot condition on impossible event
+        conditional_probability(0.5, 0)
+    except ValueError as e:
+        print(f"Error 1: {e}")
+
+    try:
+        # Intersection cannot be larger than condition
+        conditional_probability(0.7, 0.5)
+    except ValueError as e:
+        print(f"Error 2: {e}")
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ## The Conditional Paradigm
+
+        When we condition on an event, we enter a new probability universe. In this universe:
+
+        1. All probability axioms still hold
+        2. We must consistently condition on the same event
+        3. Our sample space becomes the conditioning event
+
+        Here's how our familiar probability rules look when conditioned on event $G$:
+
+        | Rule | Original | Conditioned on $G$ |
+        |------|----------|-------------------|
+        | Axiom 1 | $0 \leq P(E) \leq 1$ | $0 \leq P(E\|G) \leq 1$ |
+        | Axiom 2 | $P(S) = 1$ | $P(S\|G) = 1$ |
+        | Axiom 3* | $P(E \cup F) = P(E) + P(F)$ | $P(E \cup F\|G) = P(E\|G) + P(F\|G)$ |
+        | Complement | $P(E^C) = 1 - P(E)$ | $P(E^C\|G) = 1 - P(E\|G)$ |
+
+        *_For mutually exclusive events_
+        """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ## Multiple Conditions
+
+        We can condition on multiple events. The notation $P(E|F,G)$ means "_the probability of $E$ 
+        occurring, given that both $F$ and $G$ have occurred._"
+
+        The conditional probability formula still holds in the universe where $G$ has occurred:
+
+        $$P(E|F,G) = \frac{P(E \cap F|G)}{P(F|G)}$$
+
+        This is a powerful extension that allows us to update our probabilities as we receive 
+        multiple pieces of information.
+        """
+    )
+    return
+
+
+@app.cell
+def _():
+    def multiple_conditional_probability(p_intersection_all, p_intersection_conditions, p_condition):
+        """Calculate P(E|F,G) = P(E∩F|G)/P(F|G) = P(E∩F∩G)/P(F∩G)"""
+        if p_condition == 0:
+            raise ValueError("Cannot condition on an impossible event")
+        if p_intersection_conditions == 0:
+            raise ValueError("Cannot condition on an impossible combination of events")
+        if p_intersection_all > p_intersection_conditions:
+            raise ValueError("P(E∩F∩G) cannot be greater than P(F∩G)")
+
+        return p_intersection_all / p_intersection_conditions
+    return (multiple_conditional_probability,)
+
+
+@app.cell
+def _(multiple_conditional_probability):
+    # Example: College admissions
+    # E: Getting admitted
+    # F: High GPA
+    # G: Good test scores
+
+    # P(E∩F∩G) = P(Admitted ∩ HighGPA ∩ GoodScore) = 0.15
+    # P(F∩G) = P(HighGPA ∩ GoodScore) = 0.25
+
+    p_admit_given_both = multiple_conditional_probability(0.15, 0.25, 0.25)
+    print("College Admissions Example:")
+    print(f"P(Admitted | High GPA, Good Scores) = {p_admit_given_both}")  # Should be 0.6
+
+    # Error case: impossible condition
+    try:
+        multiple_conditional_probability(0.3, 0.2, 0.2)
+    except ValueError as e:
+        print(f"\nError case: {e}")
+    return (p_admit_given_both,)
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ## 🤔 Test Your Understanding
+
+        Which of these statements about conditional probability are true?
+
+        <details>
+        <summary>Knowing F occurred always decreases the probability of E</summary>
+        ❌ False! Conditioning on F can either increase or decrease P(E), depending on how E and F are related.
+        </details>
+
+        <details>
+        <summary>P(E|F) represents entering a new probability universe where F has occurred</summary>
+        ✅ True! We restrict ourselves to only the outcomes where F occurred, making F our new sample space.
+        </details>
+
+        <details>
+        <summary>If P(E|F) = P(E), then E and F must be the same event</summary>
+        ❌ False! This actually means E and F are independent - knowing one doesn't affect the other.
+        </details>
+
+        <details>
+        <summary>P(E|F) can be calculated by dividing P(E∩F) by P(F)</summary>
+        ✅ True! This is the fundamental definition of conditional probability.
+        </details>
+        """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        """
+        ## Summary
+
+        You've learned:
+
+        - How conditional probability updates our beliefs with new information
+        - The formula $P(E|F) = P(E \cap F)/P(F)$ and its intuition
+        - How probability rules work in conditional universes
+        - How to handle multiple conditions
+
+        In the next lesson, we'll explore **independence** - when knowing about one event 
+        tells us nothing about another.
+        """
+    )
+    return
+
+
+if __name__ == "__main__":
+    app.run()