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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.

What you'll learn

Why it matters: Card games, dice games, weather forecasts, and reliability engineering (the chance two safety systems both fail) all use compound probabilities.

Worked example

Problem. Flip a coin and roll a die. Find P(heads AND 6) as a fraction.

  1. P(heads) = 1/2 and P(6) = 1/6.
  2. Independent events multiply: (1/2)(1/6) = 1/12.

Answer: 1/12

Practice problems

1. Flip a coin twice. P(heads, heads) as a fraction.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Flip a coin twice. P(heads, heads) as a fraction.
  2. For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
  3. Each flip: P(H) = 1/2.
  4. Multiply: (1/2)(1/2) = 1/4.
  5. 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
  1. Warm-up: First identify exactly what the question is asking: Roll two dice. P(both 6) as a fraction.
  2. For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
  3. Each die: P(6) = 1/6.
  4. (1/6)(1/6) = 1/36.
  5. 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
  1. Warm-up: First identify exactly what the question is asking: Flip a coin and roll a die. P(tails AND 4) as a fraction.
  2. For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
  3. (1/2)(1/6) = 1/12.
  4. Check the result by substituting or estimating: the response should match 1/12 and make sense in the original problem.

Answer: 1/12

4. Roll two dice. P(sum equals 7) as a fraction.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Roll two dice. P(sum equals 7) as a fraction.
  2. For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
  3. 6 ways out of 36: (1,6),(2,5),(3,4),(4,3),(5,2),(6,1).
  4. 6/36 = 1/6.
  5. 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
  1. Core Practice: First identify exactly what the question is asking: Flip a coin twice. P(at least one heads) as a fraction.
  2. For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
  3. P(no heads) = (1/2)(1/2) = 1/4.
  4. P(at least one) = 1 - 1/4 = 3/4.
  5. 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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