The lesson
Read each section in order, work through every example on paper, then use the practice problems and quick check at the bottom.
Independent events
Events are independent when one doesn't affect the other (like two coin flips). For independent events, multiply the individual probabilities.
What is the probability of flipping heads and then rolling a 3?
- 1P(heads) = 1/2.
- 2P(rolling 3) = 1/6.
- 3Multiply: 1/2 × 1/6 = 1/12.
Why this matters
Find the probability of two or more events happening together.
These ideas show up on homework, quizzes, and the next unit in this grade. Getting comfortable here saves time later when the problems stack more steps together.
Common mistakes to avoid
Rushing to the answer without writing steps. Clear work helps you catch mistakes and shows your reasoning.
Re-reading the question too quickly. Underline what you are solving for before you start calculating.
Key ideas from this lesson
- Independent events
Video walkthroughs
Compound Probability of Independent Events
Multiply probabilities when events don't affect each other.
Watch on YouTube
Addition Rule for Probability
Add probabilities for mutually exclusive compound events.
Watch on YouTubePractice
For each problem: write your work in the box, type your answer, and check it. If you are stuck, reveal the solution one step at a time. Do not skip straight to the final answer.
Exercise 1
Try it yourselfFlip a fair coin and roll a fair die. Find P(heads and even number).
Step-by-step solution
- 1P(heads) = 1/2, P(even) = 3/6 = 1/2.
- 2Independent events multiply: 1/2 × 1/2 = 1/4.
Exercise 2
Try it yourselfA spinner (4 equal red, blue, green, yellow) is spun twice. Find P(red both times).
Step-by-step solution
- 1P(red) = 1/4 each spin.
- 21/4 × 1/4 = 1/16.
Exercise 3
Try it yourselfRoll two fair dice. How many equally likely outcomes are there?
Step-by-step solution
- 1Each die has 6 outcomes.
- 2Total outcomes = 6 × 6 = 36.
Exercise 4
Try it yourselfRoll two dice. Find P(sum is 7).
Step-by-step solution
- 1Favorable pairs: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) → 6 outcomes.
- 2P(sum 7) = 6/36 = 1/6.
Exercise 5
Try it yourselfDraw one card from a standard deck, replace it, then draw again. Find P(two hearts).
Step-by-step solution
- 1P(heart one draw) = 13/52 = 1/4.
- 2With replacement, draws are independent.
- 31/4 × 1/4 = 1/16.
Quick check
Answer all questions. Retake the quiz until you feel confident before moving on.
Compound Probability
Question 1 of 4
Two coins are flipped. What is the probability of getting heads on both?