# /// script # requires-python = ">=3.11" # dependencies = [ # "marimo", # "matplotlib==3.10.8", # "numpy==2.4.3", # ] # /// import marimo __generated_with = "0.18.4" app = marimo.App(width="medium") @app.cell def _(): import marimo as mo return (mo,) @app.cell(hide_code=True) def _(mo): mo.md(r""" # Axioms of Probability Probability theory is built on three fundamental axioms, known as the [Kolmogorov axioms](https://en.wikipedia.org/wiki/Probability_axioms). These axioms form the mathematical foundation for all of probability theory[¹](https://chrispiech.github.io/probabilityForComputerScientists/en/part1/probability). Let's explore each axiom and understand why they make intuitive sense: """) return @app.cell(hide_code=True) def _(mo): mo.md(r""" ## The Three Axioms | Axiom | Mathematical Form | Meaning | |-------|------------------|----------| | **Axiom 1** | $0 \leq P(E) \leq 1$ | All probabilities are between 0 and 1 | | **Axiom 2** | $P(S) = 1$ | The probability of the sample space is 1 | | **Axiom 3** | $P(E \cup F) = P(E) + P(F)$ | For mutually exclusive events, probabilities add | where the set $S$ is the sample space (all possible outcomes), and $E$ and $F$ are sets that represent events. The notation $P(E)$ denotes the probability of $E$, which you can interpret as the chance that something happens. $P(E) = 0$ means that the event cannot happen, while $P(E) = 1$ means the event will happen no matter what; $P(E) = 0.5$ means that $E$ has a 50% chance of happening. For an example, when rolling a fair six-sided die once, the sample space $S$ is the set of die faces ${1, 2, 3, 4, 5, 6}$, and there are many possible events; we'll see some examples below. """) return @app.cell(hide_code=True) def _(mo): mo.md(r""" ## Understanding Through Examples Let's explore these axioms using a simple experiment: rolling a fair six-sided die. We'll use this to demonstrate why each axiom makes intuitive sense. """) return @app.cell def _(event): event return @app.cell(hide_code=True) def _(mo): # Create an interactive widget to explore different events event = mo.ui.dropdown( options=[ "Rolling an even number (2,4,6)", "Rolling an odd number (1,3,5)", "Rolling a prime number (2,3,5)", "Rolling less than 4 (1,2,3)", "Any possible roll (1,2,3,4,5,6)", ], value="Rolling an even number (2,4,6)", label="Select an event" ) return (event,) @app.cell(hide_code=True) def _(event, mo, np, plt): # Define the probabilities for each event event_map = { "Rolling an even number (2,4,6)": [2, 4, 6], "Rolling an odd number (1,3,5)": [1, 3, 5], "Rolling a prime number (2,3,5)": [2, 3, 5], "Rolling less than 4 (1,2,3)": [1, 2, 3], "Any possible roll (1,2,3,4,5,6)": [1, 2, 3, 4, 5, 6], } # Get outcomes directly from the event value outcomes = event_map[event.value] prob = len(outcomes) / 6 # Visualize the probability dice = np.arange(1, 7) colors = ['#1f77b4' if d in outcomes else '#d3d3d3' for d in dice] fig, ax = plt.subplots(figsize=(8, 2)) ax.bar(dice, np.ones_like(dice), color=colors) ax.set_xticks(dice) ax.set_yticks([]) ax.set_title(f"P(Event) = {prob:.2f}") # Add explanation explanation = mo.md(f""" **Event**: {event.value} **Probability**: {prob:.2f} **Favorable outcomes**: {outcomes} This example demonstrates: - Axiom 1: The probability is between 0 and 1 - Axiom 2: For the sample space, P(S) = 1 - Axiom 3: The probability is the sum of individual outcome probabilities """) mo.hstack([plt.gcf(), explanation]) return @app.cell(hide_code=True) def _(mo): mo.md(r""" ## Why These Axioms Matter These axioms are more than just rules - they provide the foundation for all of probability theory: 1. **Non-negativity** (Axiom 1) makes intuitive sense: you can't have a negative number of occurrences in any experiment. 2. **Normalization** (Axiom 2) ensures that something must happen - the total probability must be 1. 3. **Additivity** (Axiom 3) lets us build complex probabilities from simple ones, but only for events that can't happen together (mutually exclusive events). From these simple rules, we can derive all the powerful tools of probability theory that are used in statistics, machine learning, and other fields. """) return @app.cell(hide_code=True) def _(mo): mo.md(r""" ## 🤔 Test Your Understanding Consider rolling two dice. Which of these statements follow from the axioms?

1. P(sum is 13) = 0

✅ Correct! This follows from Axiom 1. Since no combination of dice can sum to 13, the probability must be non-negative but can be 0.

2. P(sum is 7) + P(sum is not 7) = 1

✅ Correct! This follows from Axioms 2 and 3. These events are mutually exclusive and cover the entire sample space.

3. P(first die is 6 or second die is 6) = P(first die is 6) + P(second die is 6)

❌ Incorrect! This doesn't follow from Axiom 3 because the events are not mutually exclusive - you could roll (6,6).

""") return @app.cell def _(): import numpy as np import matplotlib.pyplot as plt return np, plt if __name__ == "__main__": app.run()