# /// script # requires-python = ">=3.10" # dependencies = [ # "marimo", # "matplotlib==3.10.8", # "matplotlib-venn==1.1.2", # "numpy==2.4.3", # ] # /// import marimo __generated_with = "0.18.4" app = marimo.App(width="medium", app_title="Conditional Probability") @app.cell def _(): import marimo as mo return (mo,) @app.cell(hide_code=True) def _(mo): mo.md(r""" # Conditional Probability _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._ 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 $F$?_"[¹](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 $P(E \mid F)$ and is defined as: $$P(E \mid F) = \frac{P(E \cap F)}{P(F)}$$ 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 start with a visual example. """) return @app.cell def _(): import matplotlib.pyplot as plt from matplotlib_venn import venn3 import numpy as np return plt, venn3 @app.cell(hide_code=True) def _(mo, plt, venn3): # Create figure with square boundaries plt.figure(figsize=(10, 3)) # 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 $F$: $$P(E \mid F) = \frac{P(E \cap F)}{P(F)} = \frac{10}{30} = \frac{1}{3} \approx 0.33$$ 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.vstack([mo.center(plt.gcf()), explanation]) return @app.cell(hide_code=True) def _(mo): mo.md(r""" Next, here's a function that computes $P(E \mid F)$, given $P( E \cap F)$ and $P(F)$ """) return @app.function 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 @app.cell def _(): # 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 @app.cell def _(): # 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 @app.cell def _(): # 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 @app.cell def _(): # 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 @app.cell def _(): # 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 | $0 \leq P(E) \leq 1$ | $0 \leq P(E \mid G) \leq 1$ | | Axiom 2 | $P(S) = 1$ | $P(S \mid G) = 1$ | | Axiom 3* | $P(E \cup F) = P(E) + P(F)$ | $P(E \cup F \mid G) = P(E \mid G) + P(F \mid G)$ | | Complement | $P(E^C) = 1 - P(E)$ | $P(E^C \mid G) = 1 - P(E \mid G)$ | *_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 $P(E \mid F,G)$ 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: $$P(E \mid F,G) = \frac{P(E \cap F \mid G)}{P(F \mid G)}$$ This is a powerful extension that allows us to update our probabilities as we receive multiple pieces of information. """) return @app.function 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 @app.cell def _(): # 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 @app.cell(hide_code=True) def _(mo): mo.md(r""" ## 🤔 Test Your Understanding Which of these statements about conditional probability are true?

Knowing F occurred always decreases the probability of E

❌ False! Conditioning on F can either increase or decrease P(E), depending on how E and F are related.

P(E|F) represents entering a new probability universe where F has occurred

✅ True! We restrict ourselves to only the outcomes where F occurred, making F our new sample space.

If P(E|F) = P(E), then E and F must be the same event

❌ False! This actually means E and F are independent - knowing one doesn't affect the other.

P(E|F) can be calculated by dividing P(E∩F) by P(F)

✅ True! This is the fundamental definition of conditional probability.

""") return @app.cell(hide_code=True) def _(mo): mo.md(r""" ## Summary You've learned: - How conditional probability updates our beliefs with new information - The formula $P(E \mid F) = P(E \cap F)/P(F)$ 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()