Compound Probability
A free Pre-Algebra lesson from the “Statistics and Probability” unit, with a worked example and practice problems including step-by-step solutions.
Two events are independent if one does not affect the other (like rolling a die and flipping a coin). For independent events, multiply: P(A and B) = P(A) * P(B). For mutually exclusive events (can't both happen), add: P(A or B) = P(A) + P(B). For 'at least one' problems, use the complement. In Statistics and Probability, the goal is not just to get an answer but to recognize the structure of the problem, choose a reliable strategy, and explain why the result is reasonable. The practice set now includes targeted skill work, transfer questions, and mixed review so students build fluency and retention.
What you'll learn
- Compute P(A and B) for independent events as P(A) * P(B)
- Compute P(A or B) for mutually exclusive events as P(A) + P(B)
- Use the complement rule P(not A) = 1 - P(A) for 'at least one' problems
Worked example
Problem. Flip a coin and roll a die. Find P(heads AND 6) as a fraction.
- P(heads) = 1/2 and P(6) = 1/6.
- Independent events multiply: (1/2)(1/6) = 1/12.
- Connect the calculation back to Compound Probability so the method, not just the arithmetic, is clear.
- Check the result against the original question before writing the final answer.
Answer: 1/12
Practice problems
1. Flip a coin twice. P(heads, heads) as a fraction.
Show solution
- Warm-up: First identify exactly what the question is asking: Flip a coin twice. P(heads, heads) as a fraction.
- For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
- Each flip: P(H) = 1/2.
- Multiply: (1/2)(1/2) = 1/4.
- Check the result by substituting or estimating: the response should match 1/4 and make sense in the original problem.
Answer: 1/4
2. Roll two dice. P(both 6) as a fraction.
Show solution
- Warm-up: First identify exactly what the question is asking: Roll two dice. P(both 6) as a fraction.
- For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
- Each die: P(6) = 1/6.
- (1/6)(1/6) = 1/36.
- Check the result by substituting or estimating: the response should match 1/36 and make sense in the original problem.
Answer: 1/36
3. Flip a coin and roll a die. P(tails AND 4) as a fraction.
Show solution
- Warm-up: First identify exactly what the question is asking: Flip a coin and roll a die. P(tails AND 4) as a fraction.
- For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
- (1/2)(1/6) = 1/12.
- Check the result by substituting or estimating: the response should match 1/12 and make sense in the original problem.
- Write the final response in the form requested by the prompt.
Answer: 1/12
4. Roll two dice. P(sum equals 7) as a fraction.
Show solution
- Core Practice: First identify exactly what the question is asking: Roll two dice. P(sum equals 7) as a fraction.
- For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
- 6 ways out of 36: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1).
- 6/36 = 1/6.
- Check the result by substituting or estimating: the response should match 1/6 and make sense in the original problem.
Answer: 1/6
5. Flip a coin twice. P(at least one heads) as a fraction.
Show solution
- Core Practice: First identify exactly what the question is asking: Flip a coin twice. P(at least one heads) as a fraction.
- For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
- P(no heads) = (1/2)(1/2) = 1/4.
- P(at least one) = 1 - 1/4 = 3/4.
- Check the result by substituting or estimating: the response should match 3/4 and make sense in the original problem.
Answer: 3/4
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