marimo-team/marimo-learn

1# /// script2# requires-python = ">=3.10"3# dependencies = [4#     "marimo",5#     "matplotlib==3.10.8",6#     "matplotlib-venn==1.1.2",7#     "numpy==2.4.3",8# ]9# ///10 11import marimo12 13__generated_with = "0.18.4"14app = marimo.App(width="medium", app_title="Conditional Probability")15 16 17@app.cell18def _():19    import marimo as mo20    return (mo,)21 22 23@app.cell(hide_code=True)24def _(mo):25    mo.md(r"""26    # Conditional Probability27 28    _This notebook is a computational companion to the book ["Probability for Computer Scientists"](https://chrispiech.github.io/probabilityForComputerScientists/en/part1/cond_prob/), by Stanford professor Chris Piech._29 30    In probability theory, we often want to update our beliefs when we receive new information.31    Conditional probability helps us formalize this process by calculating "_what is the chance of32    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/)33 34    When we condition on an event $F$:35 36    - We enter the universe where $F$ has occurred37    - Only outcomes consistent with $F$ are possible38    - Our sample space reduces to $F$39    """)40    return41 42 43@app.cell(hide_code=True)44def _(mo):45    mo.md(r"""46    ## Definition of Conditional Probability47 48    The probability of event $E$ given that event $F$ has occurred is denoted as $P(E \mid F)$ and is defined as:49 50    $$P(E \mid F) = \frac{P(E \cap F)}{P(F)}$$51 52    This formula tells us that the conditional probability is the probability of both events occurring53    divided by the probability of the conditioning event.54 55    Let's start with a visual example.56    """)57    return58 59 60@app.cell61def _():62    import matplotlib.pyplot as plt63    from matplotlib_venn import venn364    import numpy as np65    return plt, venn366 67 68@app.cell(hide_code=True)69def _(mo, plt, venn3):70    # Create figure with square boundaries71    plt.figure(figsize=(10, 3))72 73    # Draw square sample space first74    rect = plt.Rectangle((-2, -2), 4, 4, fill=False, color="gray", linestyle="--")75    plt.gca().add_patch(rect)76 77    # Set the axis limits to show the full rectangle78    plt.xlim(-2.5, 2.5)79    plt.ylim(-2.5, 2.5)80 81    # Create Venn diagram showing E and F82    # For venn3, subsets order is: (100, 010, 110, 001, 101, 011, 111)83    # Representing: (A, B, AB, C, AC, BC, ABC)84    v = venn3(subsets=(30, 20, 10, 40, 0, 0, 0), set_labels=("E", "F", "Rest"))85 86    # Customize colors87    if v:88        for id in ["100", "010", "110", "001"]:89            if v.get_patch_by_id(id):90                if id == "100":91                    v.get_patch_by_id(id).set_color("#ffcccc")  # Light red for E92                elif id == "010":93                    v.get_patch_by_id(id).set_color("#ccffcc")  # Light green for F94                elif id == "110":95                    v.get_patch_by_id(id).set_color(96                        "#e6ffe6"97                    )  # Lighter green for intersection98                elif id == "001":99                    v.get_patch_by_id(id).set_color("white")  # White for rest100 101    plt.title("Conditional Probability in Sample Space")102 103    # Remove ticks but keep the box visible104    plt.gca().set_yticks([])105    plt.gca().set_xticks([])106    plt.axis("on")107 108    # Add sample space annotation with arrow109    plt.annotate(110        "Sample Space (100)",111        xy=(-1.5, 1.5),112        xytext=(-2.2, 2),113        bbox=dict(boxstyle="round,pad=0.5", fc="white", ec="gray"),114        arrowprops=dict(arrowstyle="->"),115    )116 117    # Add explanation118    explanation = mo.md(r"""119    ### Visual Intuition120 121    In our sample space of 100 outcomes:122 123    - Event $E$ occurs in 40 cases (red region: 30 + 10)124    - Event $F$ occurs in 30 cases (green region: 20 + 10)125    - Both events occur together in 10 cases (overlap)126    - Remaining cases: 40 (to complete sample space of 100)127 128    When we condition on $F$:129    $$P(E \mid F) = \frac{P(E \cap F)}{P(F)} = \frac{10}{30} = \frac{1}{3} \approx 0.33$$130 131    This means: When we know $F$ has occurred (restricting ourselves to the green region),132    the probability of $E$ also occurring is $\frac{1}{3}$ - as 10 out of the 30 cases in the 133    green region also belong to the red region.134    """)135 136    mo.vstack([mo.center(plt.gcf()), explanation])137    return138 139 140@app.cell(hide_code=True)141def _(mo):142    mo.md(r"""143    Next, here's a function that computes $P(E \mid F)$, given $P( E \cap F)$ and $P(F)$144    """)145    return146 147 148@app.function149def conditional_probability(p_intersection, p_condition):150    if p_condition == 0:151        raise ValueError("Cannot condition on an impossible event")152    if p_intersection > p_condition:153        raise ValueError("P(E∩F) cannot be greater than P(F)")154 155    return p_intersection / p_condition156 157 158@app.cell159def _():160    # Example 1: Rolling a die161    # E: Rolling an even number (2,4,6)162    # F: Rolling a number greater than 3 (4,5,6)163    p_even_given_greater_than_3 = conditional_probability(2 / 6, 3 / 6)164    print("Example 1: Rolling a die")165    print(f"P(Even | >3) = {p_even_given_greater_than_3}")  # Should be 2/3166    return167 168 169@app.cell170def _():171    # Example 2: Cards172    # E: Drawing a Heart173    # F: Drawing a Face card (J,Q,K)174    p_heart_given_face = conditional_probability(3 / 52, 12 / 52)175    print("\nExample 2: Drawing cards")176    print(f"P(Heart | Face card) = {p_heart_given_face}")  # Should be 1/4177    return178 179 180@app.cell181def _():182    # Example 3: Student grades183    # E: Getting an A184    # F: Studying more than 3 hours185    p_a_given_study = conditional_probability(0.24, 0.40)186    print("\nExample 3: Student grades")187    print(f"P(A | Studied >3hrs) = {p_a_given_study}")  # Should be 0.6188    return189 190 191@app.cell192def _():193    # Example 4: Weather194    # E: Raining195    # F: Cloudy196    p_rain_given_cloudy = conditional_probability(0.15, 0.30)197    print("\nExample 4: Weather")198    print(f"P(Rain | Cloudy) = {p_rain_given_cloudy}")  # Should be 0.5199    return200 201 202@app.cell203def _():204    # Example 5: Error cases205    print("\nExample 5: Error cases")206    try:207        # Cannot condition on impossible event208        conditional_probability(0.5, 0)209    except ValueError as e:210        print(f"Error 1: {e}")211 212    try:213        # Intersection cannot be larger than condition214        conditional_probability(0.7, 0.5)215    except ValueError as e:216        print(f"Error 2: {e}")217    return218 219 220@app.cell(hide_code=True)221def _(mo):222    mo.md(r"""223    ## The Conditional Paradigm224 225    When we condition on an event, we enter a new probability universe. In this universe:226 227    1. All probability axioms still hold228    2. We must consistently condition on the same event229    3. Our sample space becomes the conditioning event230 231    Here's how our familiar probability rules look when conditioned on event $G$:232 233    | Rule | Original | Conditioned on $G$ |234    |------|----------|-------------------|235    | Axiom 1 | $0 \leq P(E) \leq 1$ | $0 \leq P(E \mid G) \leq 1$ |236    | Axiom 2 | $P(S) = 1$ | $P(S \mid G) = 1$ |237    | Axiom 3* | $P(E \cup F) = P(E) + P(F)$ | $P(E \cup F \mid G) = P(E \mid G) + P(F \mid G)$ |238    | Complement | $P(E^C) = 1 - P(E)$ | $P(E^C \mid G) = 1 - P(E \mid G)$ |239 240    *_For mutually exclusive events_241    """)242    return243 244 245@app.cell(hide_code=True)246def _(mo):247    mo.md(r"""248    ## Multiple Conditions249 250    We can condition on multiple events. The notation $P(E \mid F,G)$ means "_the probability of $E$251    occurring, given that both $F$ and $G$ have occurred._"252 253    The conditional probability formula still holds in the universe where $G$ has occurred:254 255    $$P(E \mid F,G) = \frac{P(E \cap F \mid G)}{P(F \mid G)}$$256 257    This is a powerful extension that allows us to update our probabilities as we receive258    multiple pieces of information.259    """)260    return261 262 263@app.function264def multiple_conditional_probability(265    p_intersection_all, p_intersection_conditions, p_condition266):267    """Calculate P(E|F,G) = P(E∩F|G)/P(F|G) = P(E∩F∩G)/P(F∩G)"""268    if p_condition == 0:269        raise ValueError("Cannot condition on an impossible event")270    if p_intersection_conditions == 0:271        raise ValueError(272            "Cannot condition on an impossible combination of events"273        )274    if p_intersection_all > p_intersection_conditions:275        raise ValueError("P(E∩F∩G) cannot be greater than P(F∩G)")276 277    return p_intersection_all / p_intersection_conditions278 279 280@app.cell281def _():282    # Example: College admissions283    # E: Getting admitted284    # F: High GPA285    # G: Good test scores286 287    # P(E∩F∩G) = P(Admitted ∩ HighGPA ∩ GoodScore) = 0.15288    # P(F∩G) = P(HighGPA ∩ GoodScore) = 0.25289 290    p_admit_given_both = multiple_conditional_probability(0.15, 0.25, 0.25)291    print("College Admissions Example:")292    print(293        f"P(Admitted | High GPA, Good Scores) = {p_admit_given_both}"294    )  # Should be 0.6295 296    # Error case: impossible condition297    try:298        multiple_conditional_probability(0.3, 0.2, 0.2)299    except ValueError as e:300        print(f"\nError case: {e}")301    return302 303 304@app.cell(hide_code=True)305def _(mo):306    mo.md(r"""307    ## 🤔 Test Your Understanding308 309    Which of these statements about conditional probability are true?310 311    <details>312    <summary>Knowing F occurred always decreases the probability of E</summary>313    ❌ False! Conditioning on F can either increase or decrease P(E), depending on how E and F are related.314    </details>315 316    <details>317    <summary>P(E|F) represents entering a new probability universe where F has occurred</summary>318    ✅ True! We restrict ourselves to only the outcomes where F occurred, making F our new sample space.319    </details>320 321    <details>322    <summary>If P(E|F) = P(E), then E and F must be the same event</summary>323    ❌ False! This actually means E and F are independent - knowing one doesn't affect the other.324    </details>325 326    <details>327    <summary>P(E|F) can be calculated by dividing P(E∩F) by P(F)</summary>328    ✅ True! This is the fundamental definition of conditional probability.329    </details>330    """)331    return332 333 334@app.cell(hide_code=True)335def _(mo):336    mo.md(r"""337    ## Summary338 339    You've learned:340 341    - How conditional probability updates our beliefs with new information342    - The formula $P(E \mid F) = P(E \cap F)/P(F)$ and its intuition343    - How probability rules work in conditional universes344    - How to handle multiple conditions345 346    In the next lesson, we'll explore **independence** - when knowing about one event347    tells us nothing about another.348    """)349    return350 351 352if __name__ == "__main__":353    app.run()354