Conditional Probability and Independence
A free Geometry lesson from the “Probability” unit, with a worked example and practice problems including step-by-step solutions.
P(B|A) — read 'probability of B given A' — is the chance B happens assuming A already happened. The formula is P(B|A) = P(A and B) / P(A). Two events are INDEPENDENT when knowing A happened does NOT change the probability of B; equivalently, P(B|A) = P(B), and P(A and B) = P(A) * P(B). If they're not independent, they're DEPENDENT.
What you'll learn
- Compute conditional probability P(B|A) = P(A and B) / P(A)
- Recognize independent events: P(A and B) = P(A) * P(B), and P(B|A) = P(B)
- Use two-way tables to find conditional probabilities
Worked example
Problem. A bag has 4 red and 6 blue marbles. P(red on second draw GIVEN red on first, without replacement)?
- After drawing one red, 3 red and 6 blue remain (9 total).
- P(red | first was red) = 3/9 = 1/3.
Answer: 3/9
Practice problems
1. Roll a fair die. P(showing 4 | result is even) as a fraction.
Show solution
- Warm-up: First identify exactly what the question is asking: Roll a fair die. P(showing 4 | result is even) as a fraction.
- For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
- Given even = sample {2, 4, 6}, three outcomes.
- Of those, 4 is one. P = 1/3.
- Check the result by substituting or estimating: the response should match 1/3 and make sense in the original problem.
Answer: 1/3
2. Roll a fair die. P(showing 6 | result is greater than 3) as a fraction.
Show solution
- Warm-up: First identify exactly what the question is asking: Roll a fair die. P(showing 6 | result is greater than 3) as a fraction.
- For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
- Greater than 3 = {4, 5, 6}, three outcomes.
- Of those, 6 is one. P = 1/3.
- Check the result by substituting or estimating: the response should match 1/3 and make sense in the original problem.
Answer: 1/3
3. If two events are independent, P(B | A) equals:
Choices: P(B) · P(A) * P(B) · P(A)
Show solution
- Warm-up: First identify exactly what the question is asking: If two events are independent, P(B | A) equals:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Independence means A does not change B's probability.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: P(B)
4. 100 students. 40 play sports; 30 are in band; 12 do both. Find P(band | sports) as a fraction.
Show solution
- Core Practice: First identify exactly what the question is asking: 100 students. 40 play sports; 30 are in band; 12 do both. Find P(band | sports) as a fraction.
- For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
- P(band AND sports) / P(sports) = 12 / 40.
- Check the result by substituting or estimating: the response should match 12/40 and make sense in the original problem.
Answer: 12/40
5. Same survey. Find P(sports | band) as a fraction.
Show solution
- Core Practice: First identify exactly what the question is asking: Same survey. Find P(sports | band) as a fraction.
- For fractions, use equivalent forms, common denominators, or reciprocals depending on the operation being used.
- P(sports AND band) / P(band) = 12 / 30.
- Check the result by substituting or estimating: the response should match 12/30 and make sense in the original problem.
Answer: 12/30
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