What is a compound event in probability?
A compound event is an event that combines two or more simple events, such as flipping heads and rolling a 6, or drawing a red card or a king. Find its probability by counting outcomes in a list, table or tree diagram, or with rules such as P(A and B) = P(A) × P(B) for independent events.
Compound Event in a picture
Formal definition and notation
A compound event is built from simpler events with and (both happen, the intersection A ∩ B) or or (at least one happens, the union A ∪ B), often over several steps of a chance process. As with any event, its probability is the fraction of equally likely outcomes in the sample space for which it occurs. Useful rules: P(A or B) = P(A) + P(B) − P(A and B), and P(A and B) = P(A) × P(B ∣ A), which becomes P(A) × P(B) when A and B are independent.
| Symbol | Read it as | Meaning |
|---|---|---|
| A ∩ B | A and B, or A intersect B | both events happen |
| A ∪ B | A or B, or A union B | at least one of the events happens |
| P(B ∣ A) | P of B given A | the probability of B once A has happened |
Key facts
- “And” usually means multiply; “or” usually means add, then subtract the overlap.
- Counting always works: list the sample space, mark the outcomes in the event, divide.
- A table suits two steps; a tree diagram suits two or more steps.
- Without replacement, the second step has fewer items, so its probability changes (dependent events).
- For mutually exclusive events there is no overlap, so P(A or B) = P(A) + P(B).
- “At least one” is often easiest with the complement: 1 − P(none).
Compound probability rules
- A,B
- the simple events being combined
- P(A and B)
- subtracted in the “or” rule so shared outcomes are not counted twice
When the steps are not independent, such as drawing without replacement, use P(A) × P(B ∣ A).
Compound Event examples, step by step
Example 1: Coin and cube
You flip a coin and roll a number cube. What is the probability of heads and an even number?
- Heads: 12. Even number: 36 = 12.
- The coin and the cube do not affect each other, so multiply: 12 × 12 = 14.
- Check by counting: H2, H4, H6 are 3 of the 12 outcomes, and 312 = 14.
Answer: The probability is 1/4.
Example 2: Two cubes, sum at least 10
You roll two number cubes. What is the probability that the sum is 10 or more?
- The sample space has 6 × 6 = 36 outcomes.
- Sum 10: (4, 6), (5, 5), (6, 4). Sum 11: (5, 6), (6, 5). Sum 12: (6, 6).
- That is 6 outcomes: P = 636 = 16.
Answer: The probability is 1/6.
Example 3: Two draws without replacement
A bag has 3 red and 2 blue marbles. You draw one marble, keep it out, and draw another. What is the probability that both are red?
- First draw: P(red) = 35.
- Now 2 red remain among 4 marbles: P(red second ∣ red first) = 24.
- Multiply: 35 × 24 = 620 = 310.
Answer: The probability that both are red is 3/10.
Common mistakes
Both things must happen, which is less likely than either one alone, so the answer must be smaller than each.
The red kings are in both events, so adding counts them twice.
Picture the cubes as two colors. Red 4 with blue 6 is not the same as red 6 with blue 4.
After a red marble leaves the bag, there are fewer reds and fewer marbles for the second draw.
Not to be confused with
A simple event is one outcome of one experiment, like rolling a 3. A compound event combines events, like rolling a 3 and then a 5.
Independence is a property of two events: one does not change the probability of the other. A compound event may involve independent or dependent events.
Quick check: 3 questions
You flip a coin and roll a number cube. What is P(tails and a 5)?
Answer: B. 12 × 16 = 112: one outcome, T5, out of 12. 23 adds the probabilities, 18 uses 2 + 6 outcomes, and 16 ignores the coin.Show answer and explanation
You roll two number cubes. What is the probability of doubles (both cubes the same)?
Answer: D. Doubles are (1, 1) through (6, 6): 6 of 36 outcomes, so 636 = 16. 136 is one specific double, like (6, 6).Show answer and explanation
A card is drawn from a standard 52-card deck. What is P(heart or face card)? (There are 13 hearts, 12 face cards, and 3 face cards are hearts.)
Answer: C. 13 + 12 − 352 = 2252. 2552 counts the 3 heart face cards twice, and 352 is “heart and face card.”Show answer and explanation
Compound Event: frequently asked questions
How do you find the probability of a compound event?
Either count or use rules. To count, list the sample space with a table or tree diagram, mark the outcomes where the compound event happens, and divide by the total. With rules, multiply for “and” (adjusting the second probability if the first step changes things) and add for “or,” subtracting any overlap.
Is a compound event the same as independent events?
No. A compound event is any combination of events, like “red on the first draw and red on the second.” Independent describes how two events relate: one does not change the other’s probability. Drawing with replacement gives independent steps; drawing without replacement gives dependent steps. Both are compound events.
Practice and tools
Sources
- Common Core State Standards, Grade 7 Statistics and Probability (7.SP.C.8)
- OpenStax, Introductory Statistics 2e, 3.1 Terminology
- OpenStax, Introductory Statistics 2e, 3.2 Independent and Mutually Exclusive Events
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 7.SP.C.8, CCSS 7.SP.C.8.a, CCSS 7.SP.C.8.b
Cite this entry
Effortless Math Team. “Compound Event.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/compound-event/
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