Statistics Addition Rule
The addition rule finds the probability that at least one of two events happens, and the exact formula you use depends on whether the events can occur together or are mutually exclusive.
P(A or B): Combining Events
When we ask for the probability of 'A or B,' we mean the probability that at least one of the two events happens -- A alone, B alone, or both. This is often called the union of the two events. How you calculate it depends on whether A and B can ever happen at the same time.
Mutually Exclusive Events
Two events are mutually exclusive if they cannot both occur in the same trial -- they share no outcomes at all. When events are mutually exclusive, the addition rule is simple: P(A or B) = P(A) + P(B), because there is no overlap to worry about double-counting.
Example: Rolling a Die
# Rolling a fair six-sided die
p_two = 1 / 6
p_five = 1 / 6
# A single roll can't show a 2 and a 5 at the same time -> mutually exclusive
p_two_or_five = p_two + p_five
print(f"P(2 or 5) = {p_two_or_five:.4f}")
# Output:
# P(2 or 5) = 0.3333Events That Can Overlap
When two events can happen at the same time, simply adding their probabilities would double-count the outcomes where both occur. The general addition rule corrects for this: P(A or B) = P(A) + P(B) - P(A and B), subtracting out the overlap exactly once.
Example: Drawing a King or a Heart
total_cards = 52
p_king = 4 / total_cards
p_heart = 13 / total_cards
p_king_and_heart = 1 / total_cards # the king of hearts is counted in both groups
p_king_or_heart = p_king + p_heart - p_king_and_heart
print(f"P(king or heart) = {p_king_or_heart:.4f}")
# Output:
# P(king or heart) = 0.3077- Ask: 'Can A and B both happen in the same trial?' If no, they are mutually exclusive.
- Rolling a 2 and rolling a 5 on one die: mutually exclusive (a single roll shows only one number).
- Drawing a king and drawing a heart from one card: not mutually exclusive (the king of hearts satisfies both).
