Probability of Compound Events
Grade 7 · Math · Study Guide
Learn how to find the probability of two or more events occurring together using sample spaces, tables, and tree diagrams.
A compound event consists of two or more simple events. When finding the probability of compound events, you need to consider whether the events are independent or dependent.
Two events are independent if the outcome of one does not affect the outcome of the other. For example, flipping a coin and rolling a die are independent. To find the probability of both independent events occurring, multiply their individual probabilities: P(A and B) = P(A) × P(B).
Two events are dependent if the outcome of one affects the probability of the other. For example, drawing two cards without replacement — once the first card is drawn, the sample space for the second changes.
A sample space is the set of all possible outcomes. For compound events, you can organize the sample space using a tree diagram or a table. For example, flipping two coins has a sample space of {HH, HT, TH, TT} — 4 total outcomes.
Probability is always written as a fraction, decimal, or percent between 0 and 1. P = (favorable outcomes) ÷ (total outcomes). The sum of all probabilities in a sample space always equals 1.
Key terms
- Compound event
- An event made up of two or more simple events
- Independent events
- Events where the outcome of one does not affect the probability of the other
- Dependent events
- Events where the outcome of one changes the probability of the other
- Sample space
- The complete set of all possible outcomes for an experiment
- Tree diagram
- A branching diagram used to list all outcomes in a sample space for compound events
- Multiplication Rule (independent)
- P(A and B) = P(A) × P(B) when events A and B are independent
Flashcards
- Simple event
- An event with only one outcome, like rolling a 4 on a die
- Compound event
- An event involving two or more outcomes, like rolling a 4 AND flipping heads
- P(A and B) for independent events
- Multiply: P(A) × P(B)
- P(A and B) for dependent events
- Multiply: P(A) × P(B|A), where P(B|A) is probability of B after A has occurred
- Tree diagram
- A visual tool showing all possible outcomes of compound events as branches
- Sample space size for compound events
- Multiply the number of outcomes of each individual event
- With replacement vs. without replacement
- With replacement: independent events (sample space stays same). Without replacement: dependent events (sample space shrinks).
- Probability formula
- P(event) = number of favorable outcomes ÷ total number of outcomes
Questions
A bag has 3 red and 2 blue marbles. You pick one, replace it, then pick again. What is P(red, then blue)?
- 1/5
- 6/25
- 6/20
- 1/4
Show answer
Answer: 6/25
P(red) = 3/5 and P(blue) = 2/5. Since the marble is replaced, these are independent. P(red then blue) = 3/5 × 2/5 = 6/25.
When you flip a fair coin twice, the probability of getting heads on the second flip is affected by the result of the first flip.
Show answer
Answer: False
Coin flips are independent events. The result of the first flip does not change the probability of the second flip — it is still 1/2.
You roll a standard die and flip a coin. The total number of outcomes in the sample space is _____.
Show answer
Answer: 12
A die has 6 outcomes and a coin has 2 outcomes. Total sample space = 6 × 2 = 12 outcomes.
A spinner has 4 equal sections: red, blue, green, yellow. You spin it twice. What is P(green, then green)?
- 1/4
- 1/2
- 1/16
- 2/4
Show answer
Answer: 1/16
P(green) = 1/4. Since the spins are independent, P(green, green) = 1/4 × 1/4 = 1/16.
Drawing a card from a deck WITHOUT replacing it before drawing a second card creates dependent events.
Show answer
Answer: True
Without replacement, after drawing the first card the deck has one fewer card, changing the probabilities for the second draw. The events are dependent.
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