The probability of “at least” something or “not” something can be found easily by using the complementary event.
Probability of a Complementary Event
P(not A) = 1 − P(A)
Example ①
When two dice are rolled, find the probability that at least one die shows 5 or more.
Solution: The complementary event, “both dice show 4 or less,” = (4/6)×(4/6) → 4×4=16 outcomes in the table. The required probability = 1−16/36 = 1−4/9 = 5/9
When two dice are rolled, find the probability that at least one die shows 5 or more.
Solution: The complementary event, “both dice show 4 or less,” = (4/6)×(4/6) → 4×4=16 outcomes in the table. The required probability = 1−16/36 = 1−4/9 = 5/9
Key Points
- For “at least one,” subtract the complementary event, “none”
- P(A) + P(complementary event of A) = 1
Practice Problems
- When a coin is tossed four times, find the probability that heads appears at least once.
Answers and Explanations
- Answer: 15/16. In all, 2⁴=16 outcomes. There is 1 outcome consisting entirely of tails. 1−1/16=15/16.