# /// script # requires-python = ">=3.10" # dependencies = [ # "marimo", # "matplotlib==3.10.8", # "matplotlib-venn==1.1.2" # ] # /// import marimo __generated_with = "0.18.4" app = marimo.App(width="medium") @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 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[¹](https://chrispiech.github.io/probabilityForComputerScientists/en/part1/prob_or/). Let's explore how to calculate $P(E \cup F)$, i.e. $P(E \text{ or } F)$, 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 @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 @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 @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?

1. P(even or less than 3) = P(even) + P(less than 3)

❌ Incorrect! These events are not mutually exclusive (2 is both even and less than 3). We need to use the inclusion-exclusion principle.

2. P(even or greater than 4) = 4/6

✅ Correct! {2,4,6} ∪ {5,6} = {2,4,5,6}, so probability is 4/6.

3. P(prime or odd) = 5/6

✅ Correct! {2,3,5} ∪ {1,3,5} = {1,2,3,5}, so probability is 5/6.

""") 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()