Probability Basics

Lesson Introduction

Welcome to our lesson on "Probability Basics"! This topic is important for making predictions and understanding uncertainties in machine learning. By the end of this lesson, you'll know the essentials of probability and how to calculate probabilities for different events using Python. Let's begin by understanding what probability is and why it matters.

What is Probability?

P

Probability of Drawing a Specific Card

P(\text{Ace}) = \frac{\text{Number of Aces}}{\text{Total cards}} = \frac{4}{52}

Probability of Drawing a Card of a Specific Suit

P(\text{Hearts}) = \frac{\text{Number of Hearts}}{\text{Total cards}} = \frac{13}{52}

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Python

# Total number of cards in a standard deck
total_cards = 52

# Number of Aces in a standard deck
aces = 4

# Probability of drawing an Ace from a standard deck
probability_of_ace = aces / total_cards
print(f"Probability of drawing an Ace: {probability_of_ace:.2f}")  # Probability of drawing an Ace: 0.08

# Total number of cards in a standard deck
total_cards = 52

# Number of Aces in a standard deck
aces = 4

# Probability of drawing an Ace from a standard deck
probability_of_ace = aces / total_cards
print(f"Probability of drawing an Ace: {probability_of_ace:.2f}")  # Probability of drawing an Ace: 0.08

Python

# Total number of cards in a standard deck
total_cards = 52

# Number of cards from the Hearts suit in a standard deck
cards_of_suit = 13

# Probability of drawing a card from the Hearts suit
probability_of_suit = cards_of_suit / total_cards
print(f"Probability of drawing a card from the Hearts suit: {probability_of_suit:.2f}")  # Probability of drawing a card from the Hearts suit: 0.25

# Total number of cards in a standard deck
total_cards = 52

# Number of cards from the Hearts suit in a standard deck
cards_of_suit = 13

# Probability of drawing a card from the Hearts suit
probability_of_suit = cards_of_suit / total_cards
print(f"Probability of drawing a card from the Hearts suit: {probability_of_suit:.2f}")  # Probability of drawing a card from the Hearts suit: 0.25

Python

# Total number of cards in a standard deck
total_cards = 52

# Number of Aces in a standard deck
aces = 4

# Number of Kings in a standard deck
kings = 4

# Probability of drawing an Ace
probability_of_ace = aces / total_cards

# Probability of drawing a King
probability_of_king = kings / total_cards

# Probability of drawing an Ace or a King (mutually exclusive)
probability_of_ace_or_king = probability_of_ace + probability_of_king
print(f"Probability of drawing an Ace or a King (mutually exclusive): {probability_of_ace_or_king:.2f}")  # Probability of drawing an Ace or a King (mutually exclusive): 0.15

# Total number of cards in a standard deck
total_cards = 52

# Number of Aces in a standard deck
aces = 4

# Number of Kings in a standard deck
kings = 4

# Probability of drawing an Ace
probability_of_ace = aces / total_cards

# Probability of drawing a King
probability_of_king = kings / total_cards

# Probability of drawing an Ace or a King (mutually exclusive)
probability_of_ace_or_king = probability_of_ace + probability_of_king
print(f"Probability of drawing an Ace or a King (mutually exclusive): {probability_of_ace_or_king:.2f}")  # Probability of drawing an Ace or a King (mutually exclusive): 0.15

Python

# Total number of cards in a standard deck
total_cards = 52

# Number of Aces in a standard deck
aces = 4

# Number of Hearts in a standard deck
hearts = 13

# Number of Ace of Hearts in a standard deck
ace_of_hearts = 1

# Probability of drawing an Ace
probability_of_ace = aces / total_cards

# Probability of drawing a Heart
probability_of_heart = hearts / total_cards

# Probability of drawing an Ace of Hearts
probability_of_ace_of_hearts = ace_of_hearts / total_cards

# Probability of drawing an Ace or a Heart (non-mutually exclusive)
probability_of_ace_or_heart = probability_of_ace + probability_of_heart - probability_of_ace_of_hearts
print(f"Probability of drawing an Ace or a Heart (non-mutually exclusive): {probability_of_ace_or_heart:.2f}")  # Probability of drawing an Ace or a Heart (non-mutually exclusive): 0.31

# Total number of cards in a standard deck
total_cards = 52

# Number of Aces in a standard deck
aces = 4

# Number of Hearts in a standard deck
hearts = 13

# Number of Ace of Hearts in a standard deck
ace_of_hearts = 1

# Probability of drawing an Ace
probability_of_ace = aces / total_cards

# Probability of drawing a Heart
probability_of_heart = hearts / total_cards

# Probability of drawing an Ace of Hearts
probability_of_ace_of_hearts = ace_of_hearts / total_cards

# Probability of drawing an Ace or a Heart (non-mutually exclusive)
probability_of_ace_or_heart = probability_of_ace + probability_of_heart - probability_of_ace_of_hearts
print(f"Probability of drawing an Ace or a Heart (non-mutually exclusive): {probability_of_ace_or_heart:.2f}")  # Probability of drawing an Ace or a Heart (non-mutually exclusive): 0.31

Python

# Total number of cards in a standard deck
total_cards = 52

# Number of Aces in a standard deck
aces = 4

# Number of Kings in a standard deck
kings = 4

# Probability of drawing an Ace
probability_of_ace = aces / total_cards

# Probability of drawing a King
probability_of_king = kings / total_cards

# Probability of drawing an Ace and then drawing a King with replacement
probability_of_ace_and_king = probability_of_ace * probability_of_king
print(f"Probability of drawing an Ace and then a King (independent, with replacement): {probability_of_ace_and_king:.4f}")  # Probability of drawing an Ace and then a King (independent, with replacement): 0.0059

# Total number of cards in a standard deck
total_cards = 52

# Number of Aces in a standard deck
aces = 4

# Number of Kings in a standard deck
kings = 4

# Probability of drawing an Ace
probability_of_ace = aces / total_cards

# Probability of drawing a King
probability_of_king = kings / total_cards

# Probability of drawing an Ace and then drawing a King with replacement
probability_of_ace_and_king = probability_of_ace * probability_of_king
print(f"Probability of drawing an Ace and then a King (independent, with replacement): {probability_of_ace_and_king:.4f}")  # Probability of drawing an Ace and then a King (independent, with replacement): 0.0059

Python

# Total number of cards in a standard deck
total_cards = 52

# Number of Aces in a standard deck
aces = 4

# Number of Kings in a standard deck
kings = 4

# Probability of drawing an Ace
probability_of_ace = aces / total_cards

# Probability of drawing a King given an Ace has been drawn
probability_of_king_given_ace_drawn = kings / (total_cards - 1)

# Probability of drawing an Ace and then drawing a King without replacement
probability_of_ace_and_king_no_replace = probability_of_ace * probability_of_king_given_ace_drawn
print(f"Probability of drawing an Ace and then a King (dependent, without replacement): {probability_of_ace_and_king_no_replace:.4f}")  # Probability of drawing an Ace and then a King (dependent, without replacement): 0.0060

# Total number of cards in a standard deck
total_cards = 52

# Number of Aces in a standard deck
aces = 4

# Number of Kings in a standard deck
kings = 4

# Probability of drawing an Ace
probability_of_ace = aces / total_cards

# Probability of drawing a King given an Ace has been drawn
probability_of_king_given_ace_drawn = kings / (total_cards - 1)

# Probability of drawing an Ace and then drawing a King without replacement
probability_of_ace_and_king_no_replace = probability_of_ace * probability_of_king_given_ace_drawn
print(f"Probability of drawing an Ace and then a King (dependent, without replacement): {probability_of_ace_and_king_no_replace:.4f}")  # Probability of drawing an Ace and then a King (dependent, without replacement): 0.0060

Python

# Total number of cards in a standard deck
total_cards = 52

# Number of Aces in a standard deck
aces = 4

# Probability of drawing an Ace from a standard deck
probability_of_ace = aces / total_cards
print(f"Probability of drawing an Ace: {probability_of_ace:.2f}")  # Probability of drawing an Ace: 0.08

Python

# Total number of cards in a standard deck
total_cards = 52

# Number of cards from the Hearts suit in a standard deck
cards_of_suit = 13

# Probability of drawing a card from the Hearts suit
probability_of_suit = cards_of_suit / total_cards
print(f"Probability of drawing a card from the Hearts suit: {probability_of_suit:.2f}")  # Probability of drawing a card from the Hearts suit: 0.25

Python

# Total number of cards in a standard deck
total_cards = 52

# Number of Aces in a standard deck
aces = 4

# Number of Kings in a standard deck
kings = 4

# Probability of drawing an Ace
probability_of_ace = aces / total_cards

# Probability of drawing a King
probability_of_king = kings / total_cards

# Probability of drawing an Ace or a King (mutually exclusive)
probability_of_ace_or_king = probability_of_ace + probability_of_king
print(f"Probability of drawing an Ace or a King (mutually exclusive): {probability_of_ace_or_king:.2f}")  # Probability of drawing an Ace or a King (mutually exclusive): 0.15

Python

# Total number of cards in a standard deck
total_cards = 52

# Number of Aces in a standard deck
aces = 4

# Number of Hearts in a standard deck
hearts = 13

# Number of Ace of Hearts in a standard deck
ace_of_hearts = 1

# Probability of drawing an Ace
probability_of_ace = aces / total_cards

# Probability of drawing a Heart
probability_of_heart = hearts / total_cards

# Probability of drawing an Ace of Hearts
probability_of_ace_of_hearts = ace_of_hearts / total_cards

# Probability of drawing an Ace or a Heart (non-mutually exclusive)
probability_of_ace_or_heart = probability_of_ace + probability_of_heart - probability_of_ace_of_hearts
print(f"Probability of drawing an Ace or a Heart (non-mutually exclusive): {probability_of_ace_or_heart:.2f}")  # Probability of drawing an Ace or a Heart (non-mutually exclusive): 0.31

Python

# Total number of cards in a standard deck
total_cards = 52

# Number of Aces in a standard deck
aces = 4

# Number of Kings in a standard deck
kings = 4

# Probability of drawing an Ace
probability_of_ace = aces / total_cards

# Probability of drawing a King
probability_of_king = kings / total_cards

# Probability of drawing an Ace and then drawing a King with replacement
probability_of_ace_and_king = probability_of_ace * probability_of_king
print(f"Probability of drawing an Ace and then a King (independent, with replacement): {probability_of_ace_and_king:.4f}")  # Probability of drawing an Ace and then a King (independent, with replacement): 0.0059

Python

# Total number of cards in a standard deck
total_cards = 52

# Number of Aces in a standard deck
aces = 4

# Number of Kings in a standard deck
kings = 4

# Probability of drawing an Ace
probability_of_ace = aces / total_cards

# Probability of drawing a King given an Ace has been drawn
probability_of_king_given_ace_drawn = kings / (total_cards - 1)

# Probability of drawing an Ace and then drawing a King without replacement
probability_of_ace_and_king_no_replace = probability_of_ace * probability_of_king_given_ace_drawn
print(f"Probability of drawing an Ace and then a King (dependent, without replacement): {probability_of_ace_and_king_no_replace:.4f}")  # Probability of drawing an Ace and then a King (dependent, without replacement): 0.0060

Lesson Introduction

What is Probability?

Probability of Drawing a Specific Card

Probability of Drawing a Card of a Specific Suit

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Probability Addition Rule (Mutually Exclusive Events)

Probability Addition Rule (Non-Mutually Exclusive Events)

Probability Multiplication Rule (Independent Events)

Probability Multiplication Rule (Dependent Events)

Lesson Summary