Compound Probability Formula
The Formula
When to use: Single-event probability asks about one thing happening. Compound probability asks about combinations: 'What's the chance of rolling a 6 AND flipping heads?' or 'What's the chance of drawing a heart OR a face card?' The word 'and' usually means multiply; the word 'or' usually means add (but subtract the overlap).
Quick Example
**OR (overlapping):** Draw a heart OR a king from a standard deck: P = \frac{13}{52} + \frac{4}{52} - \frac{1}{52} = \frac{16}{52} = \frac{4}{13}
(Subtract the king of hearts counted twice.)
Notation
What This Formula Means
The probability of two or more events occurring together (P(A \text{ and } B)) or at least one occurring (P(A \text{ or } B)), accounting for whether the events are independent or dependent.
Single-event probability asks about one thing happening. Compound probability asks about combinations: 'What's the chance of rolling a 6 AND flipping heads?' or 'What's the chance of drawing a heart OR a face card?' The word 'and' usually means multiply; the word 'or' usually means add (but subtract the overlap).
Formal View
Worked Examples
Example 1
mediumSolution
- 1 (a) Addition rule: P(A \cup B) = P(A)+P(B)-P(A \cap B) = 0.5+0.4-0.2 = 0.7
- 2 (b) Conditional probability: P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{0.2}{0.4} = 0.5
- 3 Independence check: P(A|B) = 0.5 = P(A) → A and B are independent
- 4 Verify: P(A) \times P(B) = 0.5 \times 0.4 = 0.20 = P(A \cap B) ✓
Answer
Example 2
hardCommon Mistakes
- Adding probabilities for 'and' events instead of multiplying: P(\text{heads and 6}) \neq \frac{1}{2} + \frac{1}{6}
- Forgetting to subtract the overlap in 'or' problems, leading to a probability greater than 1
- Using the simple multiplication rule P(A) \times P(B) when events are dependent—must use P(A) \times P(B|A) instead
Why This Formula Matters
Most real-world probability questions involve compound events: the chance of rain on both Saturday AND Sunday, the probability a patient has condition A OR condition B, or the likelihood of passing both exams.
Frequently Asked Questions
What is the Compound Probability formula?
The probability of two or more events occurring together (P(A \text{ and } B)) or at least one occurring (P(A \text{ or } B)), accounting for whether the events are independent or dependent.
How do you use the Compound Probability formula?
Single-event probability asks about one thing happening. Compound probability asks about combinations: 'What's the chance of rolling a 6 AND flipping heads?' or 'What's the chance of drawing a heart OR a face card?' The word 'and' usually means multiply; the word 'or' usually means add (but subtract the overlap).
What do the symbols mean in the Compound Probability formula?
P(A \cap B) for 'A and B'; P(A \cup B) for 'A or B'
Why is the Compound Probability formula important in Math?
Most real-world probability questions involve compound events: the chance of rain on both Saturday AND Sunday, the probability a patient has condition A OR condition B, or the likelihood of passing both exams.
What do students get wrong about Compound Probability?
The 'or' formula requires subtracting P(A \text{ and } B) to avoid counting the overlap twice. If events are mutually exclusive (can't both happen), the overlap is zero.
What should I learn before the Compound Probability formula?
Before studying the Compound Probability formula, you should understand: probability, independent events, conditional probability.