1. Meaning of the Addition Theorem
The addition theorem explains how to find the probability that at least one of two events happens. It is used when the situation involves the word “or”.
Examples:
- Getting a 2 or a 5 on a die.
- Drawing a heart or a face card from a deck.
The idea is to combine the chances of two events but avoid counting their common overlap twice.
2. Addition Rule for Mutually Exclusive Events
Two events are mutually exclusive when they cannot happen at the same time. Their intersection is empty.
\( P(A \cap B) = 0 \)
For such events, the addition rule becomes simple:
\( P(A \cup B) = P(A) + P(B) \)
2.1. Example
When rolling a die, the events:
- A: getting a 2
- B: getting a 5
These cannot happen together, so they are mutually exclusive.
\( P(A \cup B) = 1/6 + 1/6 = 2/6 = 1/3 \)
3. General Addition Rule (For Any Two Events)
When two events can occur together (i.e., they may overlap), we use the general addition rule.
The formula is:
\( P(A \cup B) = P(A) + P(B) - P(A \cap B) \)
The subtraction removes the overlap that was counted twice when adding \(P(A)\) and \(P(B)\).
3.1. Example
In a deck of cards:
- A: drawing a heart (13 cards)
- B: drawing a face card (12 cards)
The overlap is: heart face cards (3 cards).
So:
- \( P(A) = 13/52 \)
- \( P(B) = 12/52 \)
- \( P(A \cap B) = 3/52 \)
\( P(A \cup B) = \dfrac{13}{52} + \dfrac{12}{52} - \dfrac{3}{52} = \dfrac{22}{52} \)
4. Special Cases of the Addition Rule
The addition theorem behaves differently depending on how the events interact:
- No overlap: mutually exclusive events → simple sum
- Full overlap: one event completely inside the other → intersection equals the smaller event
- Partial overlap: use general rule
4.1. Example of Partial Overlap
Consider two events during drawing from a deck:
- A: drawing a red card (26 cards)
- B: drawing a king (4 cards)
Overlap: red kings = 2
\( P(A \cup B) = 26/52 + 4/52 - 2/52 = 28/52 \)