Introduction
Finding Probabilities of Compound Events is an important Grade 7 math skill because students are moving from simple answers toward explaining how the math works.
In this lesson, students use models, real questions, worked examples, practice problems, and two online quizzes to build confidence with finding probabilities of compound events.
What Is Finding Probabilities of Compound Events?
Finding Probabilities of Compound Events means choosing a model, naming what each number means, and explaining the strategy.
The goal is not only to get the answer. Students should be able to show the idea, explain the strategy, and check whether the answer makes sense.
Understanding Finding Probabilities of Compound Events
Before solving, students should slow down and decide what each number, shape, unit, or label represents.
- Read the question carefully and identify what is being asked.
- Choose a model, equation, table, or diagram that matches the situation.
- Solve one step at a time and keep units or labels attached.
- Use the answer explanation to check that the result makes sense.
Visual Models
For Finding Probabilities of Compound Events, a useful visual model is a quick drawing, array, table, number line, graph, or labeled diagram that shows what each number means.
Worked Examples
Example 1
Question: A fair coin is flipped twice. What is the probability of getting heads BOTH times?
- A. \(\frac{1}{8}\)
- B. \(\frac{1}{4}\)
- C. \(\frac{1}{2}\)
- D. \(\frac{3}{4}\)
- \(P(H)\cdot P(H)=\frac{1}{2}\cdot\frac{1}{2}=\frac{1}{4}\).
Answer: \(\frac{1}{4}\)
Example 2
Question: A fair six-sided die is rolled twice. What is the probability of rolling a 3 on the first roll AND a 5 on the second roll?
- A. \(\frac{1}{6}\)
- B. \(\frac{1}{12}\)
- C. \(\frac{1}{36}\)
- D. \(\frac{1}{18}\)
- \(P(3)\cdot P(5)=\frac{1}{6}\cdot\frac{1}{6}=\frac{1}{36}\) for independent events.
Answer: \(\frac{1}{36}\)
Example 3
Question: A bag contains 6 white marbles and 4 black marbles. One marble is drawn at random. What is the probability it is white?
- A. \(\frac{6}{10}\)
- B. \(\frac{3}{5}\)
- C. \(\frac{4}{10}\)
- D. \(\frac{2}{5}\)
- Total marbles: 10.
- White marbles: 6.
- Probability: \(\frac{6}{10}=\frac{3}{5}\).
Answer: \(\frac{3}{5}\)
Real-World Word Problems
Problem 1
Question: A bag contains 3 red marbles and 5 blue marbles. What is the probability of drawing a red marble first, WITHOUT replacing it, then drawing a blue marble?
- A. \(\frac{3}{8}\cdot\frac{5}{8}\)
- B. \(\frac{3}{8}\cdot\frac{4}{7}\)
- C. \(\frac{15}{56}\)
- D. \(\frac{15}{64}\)
Why it works: Without replacement: \(P(\text{red first})=\frac{3}{8}\), then \(P(\text{blue second})=\frac{5}{7}\) (only 7 left). Product: \(\frac{3}{8}\cdot\frac{5}{7}=\frac{15}{56}\).
Answer: \(\frac{15}{56}\)
Problem 2
Question: A bag has 10 marbles: 4 red, 3 blue, and 3 green. If you draw one marble, what is the probability of drawing red OR blue?
- A. \(\frac{7}{10}\)
- B. \(\frac{3}{10}\)
- C. \(\frac{4}{10}\)
- D. \(\frac{12}{100}\)
Why it works: Mutually exclusive: \(P(\text{red})+P(\text{blue})=\frac{4}{10}+\frac{3}{10}=\frac{7}{10}\).
Answer: \(\frac{7}{10}\)
Common Mistakes
- Rushing before identifying what the numbers represent.
- Choosing an operation that does not match the situation.
- Dropping labels, units, or context from the answer.
- Skipping the estimate or reasonableness check.
Strategy Tips
- Underline the question being asked.
- Use a model before jumping to computation.
- Write an equation that matches the story or picture.
- Explain the final answer in a sentence.
Practice Questions
Question 1
Two cards are drawn from a standard 52-card deck WITH replacement. What is the probability of drawing a heart on the first draw AND a spade on the second draw?
- A. \(\frac{1}{16}\)
- B. \(\frac{1}{13}\)
- C. \(\frac{1}{4}\)
- D. \(\frac{1}{8}\)
Question 2
Which of the following describes mutually exclusive events?
- A. Rolling a 3 AND rolling an even number on one die roll
- B. Spinning red AND spinning blue on one spinner
- C. Drawing a heart AND drawing a face card
- D. Flipping heads AND flipping tails on one coin flip
Question 3
A card is drawn from a standard deck. What is the probability of drawing a king OR a queen?
- A. \(\frac{4}{52}\)
- B. \(\frac{2}{13}\)
- C. \(\frac{2}{52}\)
- D. \(\frac{1}{26}\)
Question 4
A number from 1 to 20 is selected at random. What is the probability of selecting a multiple of 3 OR a multiple of 5?
- A. \(\frac{6}{20}\)
- B. \(\frac{9}{20}\)
- C. \(\frac{8}{20}\)
- D. \(\frac{10}{20}\)
Question 5
A spinner is divided into 8 equal sections numbered 1 through 8. What is the probability of spinning an odd number OR a prime number?
- A. \(\frac{4}{8}\) (only odd)
- B. \(\frac{6}{8}\) (missing overlap)
- C. \(\frac{7}{8}\)
- D. \(\frac{8}{8}\) (all outcomes)
Question 6
Consider the sample space for flipping a coin and rolling a die. How many total possible outcomes are there?
- A. 6
- B. 8
- C. 12
- D. 24
Full Answer Explanations Click to show all answers and explanations
Question 1
Answer: \(\frac{1}{16}\)
With replacement: \(P(\text{heart})\cdot P(\text{spade})=\frac{1}{4}\cdot\frac{1}{4}=\frac{1}{16}\).
Question 2
Answer: Mutually exclusive
Mutually exclusive events cannot happen at the same time. A coin cannot show both heads and tails on one flip.
Question 3
Answer: \(\frac{2}{13}\)
King OR Queen: \(P(\text{King})+P(\text{Queen})=\frac{4}{52}+\frac{4}{52}=\frac{8}{52}=\frac{2}{13}\) (mutually exclusive).
Question 4
Answer: \(\frac{9}{20}\)
Multiples of 3: {3, 6, 9, 12, 15, 18} and multiples of 5: {5, 10, 15, 20}. Total: {3, 5, 6, 9, 10, 12, 15, 18, 20} = 9 numbers. So \(\frac{9}{20}\).
Question 5
Answer: \(\frac{7}{8}\)
Odd: {1, 3, 5, 7}; Prime: {2, 3, 5, 7}. Union: {1, 2, 3, 5, 7} = 7 out of 8.
Question 6
Answer: 12 outcomes
Coin outcomes: 2 (H, T); Die outcomes: 6. Total: \(2\times 6=12\).
Connection to Standards
This lesson supports Grade 7 math expectations for reasoning, modeling, problem solving, and explaining answers clearly. It connects classroom skills to the kind of questions students see on state math assessments.
Summary
Finding Probabilities of Compound Events becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.
GOLDEN RULE
Understand the model before choosing the operation.
