Lecture 2 — Beginner tutorial

Sample space and probability measure by hand

Lecture 2 Tutorial — Sample space & probability (Beginner)

Based on Lec 02.

Goal

Feel these ideas with tiny examples:

  1. Random experiment → you choose what it is
  2. Sample space Ω → all outcomes
  3. Events → subsets of Ω
  4. Probability measure P → numbers in [0,1] with simple rules

Run cells top to bottom with Shift+Enter.

Step 0 — Imports

import numpy as np

print("Ready")

Step 1 — Choose a random experiment

Lecture point: the experiment is your design choice.

We take: one fair coin toss.

Outcomes are not numbers yet — just labels.

Omega = {"H", "T"}  # sample space
print("Sample space Ω =", Omega)
print("Number of outcomes =", len(Omega))

Step 2 — Events = subsets

Event (English) As a set
Heads {H}
Tails {T}
Anything {H, T} = Ω
Impossible empty set = ∅ 
event_heads = {"H"}
event_tails = {"T"}
event_anything = Omega
event_impossible = set()

print("Heads event:", event_heads)
print("Impossible event:", event_impossible)

Step 3 — A probability measure on this tiny space

For a fair coin:

  • P({H}) = P({T}) = 1/2
  • P(Ω) = 1
  • P(∅) = 0

Check lecture properties: non-negative, P(Ω)=1, disjoint additivity.

def P_fair_coin(event):
    """Probability of an event for a fair coin (event subset of {H,T})."""
    event = set(event)
    if not event.issubset(Omega):
        raise ValueError("Event must be a subset of Omega")
    # fair: each singleton outcome has mass 1/2
    return sum(0.5 for outcome in event)

print("P(H) =", P_fair_coin({"H"}))
print("P(T) =", P_fair_coin({"T"}))
print("P(Omega) =", P_fair_coin(Omega))
print("P(empty) =", P_fair_coin(set()))

# Disjoint additivity: H and T do not overlap
print("P(H)+P(T) =", P_fair_coin({"H"}) + P_fair_coin({"T"}))

Step 4 — Different experiment → different Ω

Same physical coin, new experiment: toss twice.

Ω₂ = {HH, HT, TH, TT}

This matches the lecture: you choose the experiment.

Omega2 = {"HH", "HT", "TH", "TT"}
print("Two-toss sample space:", Omega2)

# Event: exactly one head
exactly_one_head = {o for o in Omega2 if o.count("H") == 1}
print("Exactly one head:", exactly_one_head)

# Fair independent tosses -> each of 4 outcomes has probability 1/4
P_exact = len(exactly_one_head) / len(Omega2)
print("P(exactly one head) =", P_exact)

Step 5 — Simulate many outcomes (statistical viewpoint)

Lecture: when physics is hard, repeat observations and use statistics.

Simulate many fair tosses; the fraction of heads should approach 1/2.

rng = np.random.default_rng(0)
n = 10_000
tosses = rng.choice(["H", "T"], size=n)
frac_heads = np.mean(tosses == "H")

print(f"In {n} tosses, fraction of heads ≈ {frac_heads:.3f}")
print("True P(H) for fair coin = 0.5")

Step 6 — Mini exercises

# Exercise A: unfair coin with P(H)=0.7

def P_unfair(event, p_heads=0.7):
    event = set(event)
    mass = {"H": p_heads, "T": 1 - p_heads}
    return sum(mass[o] for o in event)

print("Unfair P(H) =", P_unfair({"H"}))
print("Unfair P(T) =", P_unfair({"T"}))
print("Unfair P(Omega) =", P_unfair({"H", "T"}))
# Exercise B: die Omega = {1,...,6}, fair — what is P(even)?

Omega_die = {1, 2, 3, 4, 5, 6}
even = {2, 4, 6}
P_even = len(even) / len(Omega_die)
print("P(even) =", P_even)

Recap

Lecture idea In this notebook
Random experiment coin once / twice / die
Sample space Ω Python set of outcomes
Event subset
Probability measure function → [0,1] with rules
Statistical view simulate many tosses

Next: random variables (turning abstract outcomes into numbers/vectors).