Kekkei - Advancing Financial Science For Everyone

Introduction

When all outcomes are equally likely, probability becomes a counting problem! This connection bridges combinatorics and probability, making complex calculations simple.

Learning Objectives:

Apply counting techniques to probability
Use uniform probability on finite spaces
Solve problems with permutations and combinations

Uniform Probability

On a finite sample space $\Omega$ with $n$ equally likely outcomes:

$P(A) = \frac{|A|}{|\Omega|} = \frac{\text{# favorable outcomes}}{\text{# total outcomes}}$

Key requirement: All outcomes must be equally likely.

Example: Fair Die

Sample space: $\Omega = \{1, 2, 3, 4, 5, 6\}$ , each with probability $1/6$ .

Event $A = \{2, 4, 6\}$ (even number):

$P(A) = \frac{|A|}{|\Omega|} = \frac{3}{6} = \frac{1}{2}$

python

import random
from scipy.special import comb

# Verify uniform probability with simulation
def simulate_uniform_die(n_trials=10000):
    outcomes = [1, 2, 3, 4, 5, 6]
    A = {2, 4, 6}  # even
    
    count = sum(1 for _ in range(n_trials) if random.choice(outcomes) in A)
    
    print(f"Theoretical P(even) = 3/6 = 0.5000")
    print(f"Simulated P(even) = {count/n_trials:.4f}")

simulate_uniform_die()

Counting with Combinations

Example: Card Hands

Draw 5 cards from 52. What's the probability of getting exactly 2 aces?

Total outcomes: $\binom{52}{5}$ (choose 5 from 52)

Favorable outcomes:

Choose 2 aces from 4: $\binom{4}{2}$
Choose 3 non-aces from 48: $\binom{48}{3}$

$P(\text{2 aces}) = \frac{\binom{4}{2} \cdot \binom{48}{3}}{\binom{52}{5}}$

python

from scipy.special import comb

def prob_two_aces():
    # Favorable outcomes
    ways_2_aces = int(comb(4, 2, exact=True))
    ways_3_non_aces = int(comb(48, 3, exact=True))
    favorable = ways_2_aces * ways_3_non_aces
    
    # Total outcomes
    total = int(comb(52, 5, exact=True))
    
    probability = favorable / total
    
    print(f"Ways to choose 2 aces: C(4,2) = {ways_2_aces}")
    print(f"Ways to choose 3 non-aces: C(48,3) = {ways_3_non_aces:,}")
    print(f"Favorable outcomes: {favorable:,}")
    print(f"Total outcomes: C(52,5) = {total:,}")
    print(f"\nP(exactly 2 aces) = {probability:.6f} ≈ {probability:.2%}")

prob_two_aces()

Key Takeaways

Uniform probability: $P(A) = |A| / |\Omega|$ when outcomes are equally likely
Combinatorics + Probability: Use counting to compute probabilities
Pattern: Count favorable outcomes, divide by total outcomes

Next Module: Conditional Probability and Independence!

IntroductionFocusStart Focus Mode

Uniform ProbabilityFocusStart Focus Mode