Practice Questions
Concept Explanation
At least one means one or more successes. The complement method is usually fastest: subtract P(none) from 1.
Key Formula
- P(at least one) = 1 − P(none)
- For independent trials: P(at least one) = 1 − P(fail)^n
- Works for coins, dice rolls, and repeated draws with replacement
Example
Three coins: P(at least one head) = 1 − (1/2)^3 = 7/8.
Common Trap
At least one is not the same as exactly one.
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Q1 Easy
Two fair coins. What is P(at least one head)?
Explanation1 − 1/4 = 3/4. -
Q2 Easy
Three fair coins. What is P(at least one head)?
Explanation1 − (1/2)^3 = 7/8. -
Q3 Easy
One fair die rolled. What is P(at least one 6) in one roll?
ExplanationSingle roll: P = 1/6. -
Q4 Medium
Two fair dice. What is P(at least one shows 6)?
Explanation1 − (5/6)^2 = 11/36. -
Q5 Medium
Four fair coins. What is P(at least one tail)?
Explanation1 − (1/2)^4 = 15/16. -
Q6 Medium
P(single shot hit) = 0.2. Two independent shots. P(at least one hit)?
Explanation1 − 0.8^2 = 1 − 0.64 = 0.36. -
Q7 Easy
Three fair coins. What is P(at least one tail)?
Explanation1 − P(all heads) = 7/8.
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Q8 Hard
Bag: P(red) = 0.3 each draw with replacement. Two draws. P(at least one red)?
Explanation1 − 0.7^2 = 1 − 0.49 = 0.51. -
Q9 Medium
Two fair dice. What is P(at least one die shows 1)?
Explanation1 − (5/6)^2 = 11/36. -
Q10 Hard
Five fair coins. What is P(at least one head)?
Explanation1 − (1/2)^5 = 31/32. -
Q11 Hard
P(rain each day) = 0.1, independent days. Two days. P(at least one rainy day)?
Explanation1 − 0.9^2 = 0.19. -
Q12 Hard
Three fair dice. What is P(at least one shows 6)?
Explanation1 − (5/6)^3 = 1 − 125/216 = 91/216. -
Q13 Easy
Two fair coins. What is P(at least one tail)?
ExplanationSame as at least one head: 3/4. -
Q14 Medium
Die rolled twice. P(at least one roll is 6)?
Explanation1 − (5/6)^2 = 11/36.
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DS1 Easy
What is P(at least one head when two coins are tossed)?
Statement ITwo fair coins are tossed.
Statement IIP(both tails) = 1/4.
ExplanationStatement II alone: P = 1 − 1/4 = 3/4. -
DS2 Medium
What is P(at least one head when three coins are tossed)?
Statement IThree fair coins are tossed.
Statement IIP(all tails) = 1/8.
ExplanationStatement II alone: P = 1 − 1/8 = 7/8.
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DS3 Medium
What is P(at least one 6 when two dice are rolled)?
Statement ITwo fair dice are rolled.
Statement IIP(no six on either die) = 25/36.
ExplanationStatement II alone: P = 11/36. -
DS4 Hard
What is P(at least one hit in two independent shots)?
Statement IEach shot has hit probability 0.3.
Statement IIThe shots are independent.
ExplanationNeed both: P = 1 − 0.7^2 = 0.51.