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Compound Event
Probability

Learn how to find the probability of compound events by counting favorable outcomes in the sample space.

7th Grade Math

What Is a Simple Event?

A simple event has only one outcome. When you flip a single coin, there are exactly two simple events: Heads or Tails.

H Heads T Tails

What Is a Compound Event?

A compound event combines two or more simple events. For example: flipping a coin and rolling a die. The result is a pair like (Heads, 3).

The Sample Space

The sample space is the set of all possible outcomes. For a compound event, it includes every combination of outcomes from each simple event.

Coin Die: 1 2 3 4 5 6 H (H,1) (H,2) (H,3) (H,4) (H,5) (H,6) T (T,1) (T,2) (T,3) (T,4) (T,5) (T,6)
Total outcomes: \(2 \times 6 = 12\)
Highlighted: (H, 3) is one outcome of 12

The Core Formula

The probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs:

This is the same idea as simple probability, but applied to combinations of events.

Counting with a Tree Diagram

A tree diagram shows every possible outcome by branching at each stage. Each path from root to leaf is one outcome.

Start H T (H, 1) (H, 2) (H, 3) ★ (H, 4) (H, 5) (H, 6) (T, 1) (T, 2) (T, 3) (T, 4) (T, 5) (T, 6) ★ = favorable 12 total paths

Simple vs Compound Events

Simple Event Compound Event
Definition One action or trial Two or more actions combined
Example Roll a die once Flip a coin and roll a die
Sample Space {1, 2, 3, 4, 5, 6} {(H,1), (H,2), ..., (T,6)}
Size 6 outcomes \(2 \times 6 = 12\) outcomes
Tools Simple list Tree, table, or organized list

How to Solve Step by Step

1
Identify the events. What simple events are being combined?
2
List the sample space. Use a tree diagram, table, or organized list to find all possible outcomes.
3
Count favorable outcomes. Which outcomes match the event you want?
4
Write the fraction. \(P = \frac{\text{favorable}}{\text{total}}\). Simplify if possible.

Example 1: Coin + Die

You flip a coin and roll a standard die. What is the probability of getting Heads and an even number?

Visualizing Example 1

Coin 1 2 3 4 5 6 H (H,1) (H,2) (H,3) (H,4) (H,5) (H,6) T (T,1) (T,2) (T,3) (T,4) (T,5) (T,6) 3 favorable out of 12 total \(P = \frac{3}{12} = \frac{1}{4}\)

Example 2: Two Spinners

Spinner A has 3 equal sections: Red, Blue, Green. Spinner B has 2 equal sections: 1, 2. What is the probability of spinning Blue and 2?

Red Blue Green Spinner A 1 2 Spinner B

Example 3: Two Cards

A bag has cards labeled A, B, C. You draw one card, put it back, and draw again. What is the probability of getting the same letter both times?

1st Draw A B C 2nd A (A,A) (B,A) (C,A) B (A,B) (B,B) (C,B) C (A,C) (B,C) (C,C) 3 favorable out of 9 total

The Counting Principle

Instead of listing every outcome, you can multiply the number of outcomes for each event to find the total sample space size:

Coin + Die
\(2 \times 6 = 12\)
2 Spinners
\(3 \times 2 = 6\)
2 Cards
\(3 \times 3 = 9\)

Why the Sample Space Matters

If you miss even one outcome, your probability will be wrong. Organized lists, tables, and tree diagrams help you find every outcome without repeating or skipping any.

1
Tree diagrams work best when events happen in a sequence (first this, then that).
2
Tables (grids) work best when there are exactly two events to combine.
3
Organized lists work when events have few outcomes and you can list systematically.

Common Mistakes

✕ Missing Outcomes
Forgetting to list all combinations. For example, listing (H,1) through (H,6) but forgetting all the (T,...) outcomes.
✕ Adding Instead of Multiplying
Writing \(2 + 6 = 8\) instead of \(2 \times 6 = 12\) for total outcomes in a coin-and-die problem.
✕ Not Simplifying
Leaving the answer as \(\frac{3}{12}\) when the simplified form \(\frac{1}{4}\) is expected.

Summary Table

Concept Key Idea
Sample Space All possible outcomes of the compound event
Total Outcomes Multiply the number of outcomes for each event
Favorable Outcomes Outcomes where the desired event occurs
Probability Formula \(P = \frac{\text{favorable}}{\text{total}}\)
Best Tools Tree diagrams, tables, organized lists

Question 1 of 5

You flip a coin and roll a 6-sided die. How many outcomes are in the sample space?

Question 2 of 5

You flip a coin and roll a die. What is the probability of getting Tails and a number greater than 4?

Question 3 of 5

A spinner has equal sections Red, Blue, Yellow. You spin it twice. What is the probability of getting Red both times?

Question 4 of 5

You roll two standard dice. How many total outcomes are in the sample space?

Question 5 of 5

You roll two dice. What is the probability that both dice show the same number (doubles)?