How to Solve Probability Problems? (+FREE Worksheet!)
Probability measures how likely an event is to occur, expressed as a number between 0 and 1 (or 0% and 100%). Understanding how to set up and solve probability problems is a key skill in Algebra 1 statistics. This guide covers the basic probability formula, simple and compound events, and includes worked examples, two video lessons, and practice problems so you can master the concept.
What Is Probability?
The probability of an event is the ratio of the number of favorable outcomes to the total number of equally likely outcomes. A probability of 0 means the event is impossible; a probability of 1 means the event is certain. All other probabilities fall strictly between 0 and 1.
\(\color{blue}{P(\text{ event }) = (\text{ Number of favorable outcomes })}\) ÷ (Total number of possible outcomes)
Types of Probability Events
Simple Events
A simple event has a single outcome. Apply the basic formula directly.
Quick example: Rolling a standard 6-sided die, what is the probability of rolling a 4?
\(\color{blue}{P(4) = 1}\) ÷ 6
Compound Events — “And” (Intersection)
For independent events (the result of one does not affect the other), multiply the probabilities:
\(\color{blue}{P(A \text{ and B }) = P(A) \times P(B)}\)
Quick example: Flipping a fair coin twice: \(\color{blue}{P(\text{ Heads and Heads }) = (\frac{1}{2}) \times (\frac{1}{2}) = \frac{1}{4}}\)
Compound Events — “Or” (Union)
For mutually exclusive events (they cannot both happen at the same time), add the probabilities:
\(\color{blue}{P(A \text{ or B }) = P(A) + P(B)}\)
Quick example: Rolling a 2 or a 5 on a die: \(\color{blue}{P(2 \text{ or } 5) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}}\)
Complementary Events
The complement of event A is everything that is NOT A. The two probabilities must sum to 1:
\(\color{blue}{P(\text{ not A }) = 1 – P(A)}\)
Quick example: \(\color{blue}{P(\text{ not rolling a } 3) = 1 – \frac{1}{6} = \frac{5}{6}}\)
Step-by-Step Summary
- Identify the total number of equally likely outcomes in the sample space.
- Count the number of favorable outcomes for the event.
- Apply the formula: \(\color{blue}{P(\text{ event }) = \text{ favorable }}\) ÷ total.
- For compound events: multiply probabilities for “and” (independent); add for “or” (mutually exclusive).
- Use \(\color{blue}{P(\text{ not A }) = 1 – P(A)}\) for complementary events.
- Simplify the fraction and, if needed, convert to a decimal or percentage.
Watch: Basic Probability (Video Lesson)
Math Antics introduces probability with clear, visual explanations:
Probability Problems – Worked Examples
Example 1: A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. A marble is drawn at random. Find the probability of drawing a red marble.
Total \(\color{blue}{\text{ outcomes } = 5 + 3 + 2 = 10}\)
Favorable outcomes \(\color{blue}{(\text{ red }) = 5}\)
\(\color{blue}{P(\text{ red }) = 5}\) ÷ 10 = \(\color{blue}{\frac{1}{2}}\)
Example 2: Using the same bag, find the probability of NOT drawing a blue marble.
\(\color{blue}{P(\text{ blue }) = 3}\) ÷ 10
\(\color{blue}{P(\text{ not blue }) = 1 – \frac{3}{10}}\) = \(\color{blue}{\frac{7}{10}}\)
Example 3: A fair coin is flipped twice. What is the probability of getting exactly one head?
Sample space: HH, HT, TH, TT → 4 equally likely outcomes
Favorable outcomes (exactly one head): HT, TH → 2 outcomes
\(\color{blue}{P(\text{ exactly one head }) = 2}\) ÷ 4 = \(\color{blue}{\frac{1}{2}}\)
Example 4: A 6-sided die is rolled. What is the probability of rolling an even number or a number greater than 4?
Even numbers: {2, 4, 6}; Numbers > 4: {5, 6}. Union: {2, 4, 5, 6} → 4 outcomes
\(\color{blue}{P(\text{ even or } > 4) = 4}\) ÷ 6 = \(\color{blue}{\frac{2}{3}}\)
More Practice: Independent and Dependent Events (Video Lesson)
Khan Academy explains independent and dependent events with detailed examples:
Exercises: Probability Problems
- A bag holds 4 yellow, 6 orange, and 10 purple chips. What is P(yellow)?
- A spinner has 8 equal sections numbered 1–8. What is P(prime number)?
- A fair coin is flipped and a die is rolled. What is P(heads and 6)?
- A deck of 52 cards is shuffled. What is P(drawing a king or a queen)?
- The probability of rain on Saturday is 0.3. What is the probability it does NOT rain?
- Two dice are rolled. What is P(\(\color{blue}{\text{ sum } = 7}\))?
Answers
- 4 ÷ 20 = \(\color{blue}{\frac{1}{5}}\)
- Prime numbers 1–8: {2, 3, 5, 7} → 4 ÷ 8 = \(\color{blue}{\frac{1}{2}}\)
- \(\color{blue}{P(\text{ heads }) \times P(6) = \frac{1}{2} \times \frac{1}{6}}\) = \(\color{blue}{\frac{1}{12}}\)
- \(\color{blue}{P(\text{ king }) + P(\text{ queen }) = \frac{4}{52} + \frac{4}{52} = \frac{8}{52}}\) = \(\color{blue}{\frac{2}{13}}\)
- \(\color{blue}{1 – 0.3}\) = 0.7
- Pairs summing to 7: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1) → 6 ÷ 36 = \(\color{blue}{\frac{1}{6}}\)
Free Probability Problems Worksheet
Ready to practice on your own? Download our free Probability Problems worksheet below, work through each problem at your own pace, and then check your answers. If a few give you trouble, scroll back up to the worked examples and try again — steady practice is the surest way to master Probability Problems before a quiz or test.
Download Probability Worksheet
Frequently Asked Questions
What is the difference between theoretical and experimental probability?
Theoretical probability is calculated using known outcomes (e.g., \(\color{blue}{\frac{1}{6}}\) for rolling a specific number on a fair die). Experimental probability is determined by actually performing the experiment many times and recording results: \(\color{blue}{P(\text{ event }) = (\text{ number of times event occurred })}\) ÷ (total trials). As the number of trials increases, experimental probability approaches theoretical probability.
What does it mean for two events to be independent?
Two events are independent if the outcome of one has no effect on the outcome of the other. For example, flipping a coin and rolling a die are independent because the coin result does not change the die result. For independent events, \(\color{blue}{P(A \text{ and B }) = P(A) \times P(B)}\).
Can a probability be greater than 1?
No. Probability is always between 0 and 1 (inclusive). A probability of 0 means the event never occurs; a probability of 1 means it always occurs. Any calculation that gives a value outside [0, 1] contains an error.
Related Topics
Related to This Article
More math articles
- Algebra Puzzle – Challenge 53
- Geometry Puzzle – Challenge 60
- TASC Testing Accommodation and Support for Disabilities
- The Ultimate 7th Grade PSSA Math Course (+FREE Worksheets)
- 5th Grade NYSE Math FREE Sample Practice Questions
- Sentence Solvers: How to Completing Addition and Subtraction Equations
- Exponential Growth and Decay Worksheet with Answers
- The Best Grade 8 Math Book for Tennessee Students
- 2nd Grade Mathematics Worksheets: FREE & Printable
- How to Identify Expressions and Equations?
What people say about "How to Solve Probability Problems? (+FREE Worksheet!) - Effortless Math: We Help Students Learn to LOVE Mathematics"?
No one replied yet.