Sample Spaces and Events¶

Overview¶

Probability theory is built upon a set of fundamental rules known as Kolmogorov's Axioms, named after the Russian mathematician Andrey Kolmogorov. These axioms provide a formal foundation for reasoning about probability and ensure consistency when calculating the likelihood of events.

Basic Definitions¶

Sample¶

A possible outcome \(\omega\) of an experiment is called a sample.

Sample Space¶

The sample space \(\Omega\) is the set of all possible outcomes (samples) of an experiment:

\[ \Omega = \{\omega_1, \omega_2, \omega_3, \ldots\} \]

Event¶

An event \(A\) is any subset of \(\Omega\). It represents a collection of outcomes of interest.

\[ A \subseteq \Omega \]

Intuitive Picture: Bricks and Weights¶

For each outcome \(\omega \in \Omega\), we attach a "brick" with a certain weight representing its probability. Different bricks may have different weights, but the total weight of all bricks across the sample space is 1. This weight distribution over \(\Omega\) defines a probability measure:

\[ \begin{aligned} P(\omega) &= \text{Weight of the brick attached to } \omega \\ P(A) &= \sum_{\omega \in A} P(\omega) = \text{Total weight of the bricks attached to } A \end{aligned} \]

Examples¶

Example: Rolling a Six-Sided Die¶

When rolling a fair six-sided die:

Sample space: \(\Omega = \{1, 2, 3, 4, 5, 6\}\)
Event \(A\) (rolling an even number): \(A = \{2, 4, 6\}\)
Event \(B\) (rolling an odd number): \(B = \{1, 3, 5\}\)

Since the die is fair, each outcome has equal probability \(P(\omega) = \frac{1}{6}\).

Example: Flipping Three Coins¶

When flipping three coins:

Sample space: \(\Omega = \{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\}\)
Event (exactly 2 heads): \(A = \{HHT, HTH, THH\}\)
\(P(A) = \frac{3}{8}\)

Python Exploration¶

from itertools import product

# Sample space for three coin flips
sample_space = list(product(['H', 'T'], repeat=3))
print(f"Sample space size: {len(sample_space)}")
print(f"Sample space: {sample_space}")

# Event: exactly 2 heads
event_2_heads = [s for s in sample_space if s.count('H') == 2]
print(f"\nEvent (2 heads): {event_2_heads}")
print(f"P(2 heads) = {len(event_2_heads)}/{len(sample_space)} = {len(event_2_heads)/len(sample_space):.4f}")

Key Takeaways¶

The sample space \(\Omega\) captures every possible outcome of an experiment.
An event is any subset of \(\Omega\).
Probability assigns a non-negative "weight" to each outcome such that the total weight is 1.
The probability of an event is the sum of weights of all outcomes in that event.