The 'at least one' trick

Goal: Find P(at least one) as 1 − P(none).

Remember: It is usually easier to count the hands with none, then subtract the probability from 1.

1. 1.How many five-card hands contain no ace?

   Answer: 1,712,304

   Working: C(48, 5)

2. 2.What is the probability a five-card hand contains no ace? Give a percentage to 2 decimal places.

   Answer: ≈ 65.88%

   Working: 1,712,304 ÷ 2,598,960

3. 3.What is the probability a five-card hand contains at least one ace?

   Answer: ≈ 34.12%

   Working: 1 − 0.6588

4. 4.Two cards are drawn. What is the probability at least one is a heart?

   Answer: ≈ 44.12%

   Working: 1 − (39/52 × 38/51) = 1 − 1,482/2,652

5. 5.Two cards are drawn. What is the probability at least one is red?

   Answer: ≈ 75.49%

   Working: 1 − (26/52 × 25/51) = 1 − 650/2,652

6. 6.Explain in a sentence why counting 'none' is easier than counting 'at least one'.

   Answer: 'At least one' covers many cases (exactly 1, 2, 3 ...), whereas 'none' is a single case; subtracting it from 1 gives all the others.