The Multiplication Rule of Probability

We have already considered that if events A and B are independent, then:
P(A and B) = P(A) *P(B).
This formula is a special case of the more general probabilities multiplication rule:

It states that the probability of the joint occurrence of two events, A and B, is equal to the probability of event A multiplied by event B's conditional probability, given that event A has occurred.

Example

Assume you draw two cards from a standard deck (52 cards) without replacement. What is the probability of drawing a heart on the first card and a diamond on the second?
Event A - drawing a heart first. Event B - drawing a diamond second.

import numpy as np

# Creating a deck of 52 cards

suits = ['H', 'D', 'C', 'S'] # Hearts, Diamonds, Clubs, Spades

ranks = ['2', '3', '4', '5', '6', '7', '8', '9', '10', 'J', 'Q', 'K', 'A']

deck = [rank + suit for suit in suits for rank in ranks]

# Counting the number of cards in the deck

total_cards = len(deck)

# Counting the number of hearts and diamonds in the deck

hearts = sum(card[-1] == 'H' for card in deck)

diamonds = sum(card[-1] == 'D' for card in deck)

# Calculating P(A)

p_A = hearts / total_cards

# Calculating P(B|A)

# We have already removed one heart from the deck

# Total number of cards has become 1 less

# As a result conditional probability can be calculated

p_B_cond_A = diamonds / (total_cards - 1)

# Resulting probability due to multiplication rule

p = p_A * p_B_cond_A

print(f'Resulting probability is {p:.4f}')


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import numpy as np

# Creating a deck of 52 cards
suits = ['H', 'D', 'C', 'S']  # Hearts, Diamonds, Clubs, Spades
ranks = ['2', '3', '4', '5', '6', '7', '8', '9', '10', 'J', 'Q', 'K', 'A']
deck = [rank + suit for suit in suits for rank in ranks]

# Counting the number of cards in the deck
total_cards = len(deck)

# Counting the number of hearts and diamonds in the deck
hearts = sum(card[-1] == 'H' for card in deck)
diamonds = sum(card[-1] == 'D' for card in deck)

# Calculating P(A)
p_A = hearts / total_cards

# Calculating P(B|A) 
# We have already removed one heart from the deck
# Total number of cards has become 1 less
# As a result conditional probability can be calculated
p_B_cond_A = diamonds / (total_cards - 1)

# Resulting probability due to multiplication rule
p = p_A * p_B_cond_A
print(f'Resulting probability is {p:.4f}')

Note

Pay attention that in the multiplication probabilities rule, the order in which events occur is unimportant - we can consider both the probability P(B)*P(A|B) and P(A)*P(B|A).

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Sezione 2. Capitolo 3