Statistics Multiplication Rule
The multiplication rule finds the probability that two events both happen, and it takes a different form depending on whether the events are independent or one depends on the other.
P(A and B): Both Events Happening
When we ask for the probability of 'A and B,' we mean the probability that both events happen together -- this is called the intersection of the two events. The formula you need depends on whether knowing that A happened changes the probability of B.
Independent Events
Two events are independent if the outcome of one has no effect whatsoever on the probability of the other. When events are independent, the multiplication rule is simple: P(A and B) = P(A) x P(B).
Example: A Coin Flip and a Die Roll
p_heads = 1 / 2
p_six = 1 / 6
# The coin flip has no effect on the die roll -> independent events
p_heads_and_six = p_heads * p_six
print(f"P(heads and rolling a 6) = {p_heads_and_six:.4f}")
# Output:
# P(heads and rolling a 6) = 0.0833Dependent Events and Conditional Probability
Two events are dependent if the outcome of one changes the probability of the other. In that case we need conditional probability, written P(B|A), meaning 'the probability of B, given that A already happened.' The general multiplication rule becomes: P(A and B) = P(A) x P(B|A).
Example: Drawing Two Aces Without Replacement
p_first_ace = 4 / 52
p_second_ace_given_first = 3 / 51 # one ace and one card are already gone from the deck
p_both_aces = p_first_ace * p_second_ace_given_first
print(f"P(first ace) = {p_first_ace:.4f}")
print(f"P(second ace | first ace) = {p_second_ace_given_first:.4f}")
print(f"P(both aces) = {p_both_aces:.4f}")
# Output:
# P(first ace) = 0.0769
# P(second ace | first ace) = 0.0588
# P(both aces) = 0.0045- Ask: 'Does knowing A happened change the probability of B?' If no, the events are independent.
- Drawing with replacement (returning the card or marble before the next draw) keeps each draw independent, since the odds reset every time.
- Drawing without replacement makes each draw dependent, since removing an item changes what's left in the pool.
