Probability Basics

What is Probability?

Probability measures how likely something is to happen.

0 = impossible
1 = certain
0.5 = 50% chance

Basic Formula

Probability = Favorable outcomes / Total outcomes

Simple Examples

Coin Flip

code.py

# Probability of heads
favorable = 1  # heads
total = 2      # heads or tails

p_heads = favorable / total
print(p_heads)  # 0.5

Dice Roll

code.py

# Probability of rolling a 6
favorable = 1  # only one 6
total = 6      # numbers 1-6

p_six = favorable / total
print(p_six)  # 0.167

Simulating Probability

code.py

import numpy as np

# Simulate 10000 coin flips
flips = np.random.choice(['heads', 'tails'], size=10000)

# Count heads
heads_count = np.sum(flips == 'heads')
probability = heads_count / 10000

print(f"Probability of heads: {probability}")  # ~0.5

Complement Rule

Probability something does NOT happen:

P(not A) = 1 - P(A)

code.py

p_rain = 0.3
p_no_rain = 1 - p_rain
print(p_no_rain)  # 0.7

Addition Rule (OR)

Probability of A or B happening:

code.py

# Probability of rolling 1 OR 2
p_one = 1/6
p_two = 1/6

# If events can't happen together (mutually exclusive)
p_one_or_two = p_one + p_two
print(p_one_or_two)  # 0.333

Multiplication Rule (AND)

Probability of A and B happening:

code.py

# Probability of two heads in a row
p_heads = 0.5

# If events are independent
p_two_heads = p_heads * p_heads
print(p_two_heads)  # 0.25

Independent Events

Events that don't affect each other:

code.py

# Each coin flip is independent
p_heads = 0.5

# 5 heads in a row
p_five_heads = p_heads ** 5
print(p_five_heads)  # 0.03125

Conditional Probability

Probability of A given B happened:

P(A|B) = P(A and B) / P(B)

code.py

# 60% of customers are female
# 40% of females buy product
# What's P(buy | female)?

p_female = 0.60
p_buy_and_female = 0.24

p_buy_given_female = p_buy_and_female / p_female
print(p_buy_given_female)  # 0.4

Bayes' Theorem

P(A|B) = P(B|A) * P(A) / P(B)

code.py

# Disease test example
# P(disease) = 0.01 (1% have disease)
# P(positive | disease) = 0.99 (99% accuracy)
# P(positive | no disease) = 0.05 (5% false positive)

p_disease = 0.01
p_positive_given_disease = 0.99
p_positive_given_no_disease = 0.05

# P(positive)
p_positive = (p_positive_given_disease * p_disease +
              p_positive_given_no_disease * (1 - p_disease))

# P(disease | positive)
p_disease_given_positive = (p_positive_given_disease * p_disease) / p_positive
print(f"P(disease|positive): {p_disease_given_positive:.2%}")  # ~16.7%

Even with 99% accurate test, only 17% chance of actually having disease!

Expected Value

Average outcome over many tries:

code.py

# Dice roll expected value
outcomes = [1, 2, 3, 4, 5, 6]
probability = 1/6

expected = sum(outcome * probability for outcome in outcomes)
print(expected)  # 3.5

Complete Example

code.py

import numpy as np

# Simulate dice rolls
np.random.seed(42)
rolls = np.random.randint(1, 7, size=10000)

# Calculate probabilities
print("=== Dice Roll Probabilities ===")
for i in range(1, 7):
    prob = np.sum(rolls == i) / len(rolls)
    print(f"P({i}): {prob:.3f}")

# P(even number)
p_even = np.sum(rolls % 2 == 0) / len(rolls)
print(f"P(even): {p_even:.3f}")

# P(greater than 4)
p_gt_4 = np.sum(rolls > 4) / len(rolls)
print(f"P(>4): {p_gt_4:.3f}")

Key Points

Probability is between 0 and 1
P(not A) = 1 - P(A)
P(A or B) = P(A) + P(B) (if mutually exclusive)
P(A and B) = P(A) × P(B) (if independent)
Conditional probability considers given info
Bayes' theorem updates probability with new evidence

What's Next?

Learn about probability distributions.