Compound Events Statistics Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

hard
A deck has 52 cards. You draw 2 cards without replacement. What is the probability of drawing at least one ace?

Solution

  1. 1
    Step 1: Use the complement: P(at least one ace)=1−P(no aces)P(\text{at least one ace}) = 1 - P(\text{no aces}).
  2. 2
    Step 2: P(no aces)=4852×4751=22562652=188221≈0.8507P(\text{no aces}) = \frac{48}{52} \times \frac{47}{51} = \frac{2256}{2652} = \frac{188}{221} \approx 0.8507. So P(at least one ace)=1−0.8507=0.1493P(\text{at least one ace}) = 1 - 0.8507 = 0.1493.

Answer

P(at least one ace)=1−188221=33221≈0.149P(\text{at least one ace}) = 1 - \frac{188}{221} = \frac{33}{221} \approx 0.149 (about 14.9%).
The complement method is efficient for 'at least one' problems. Instead of calculating all the ways to get one or two aces separately, we find the probability of the complement (no aces) and subtract from 1. This technique is widely used in probability.

About Compound Events

Compound events are probability events made up of two or more simple events combined using 'and' (both events occur) or 'or' (at least one occurs). For independent 'and' events, multiply probabilities; for 'or' events, add probabilities and subtract any overlap.

Learn more about Compound Events →

More Compound Events Examples

Example 1 easy

A bag contains 3 red and 5 blue marbles. You draw one marble, replace it, then draw another. What is

Example 2 medium

A bag contains 4 red and 6 blue marbles. You draw two marbles without replacement. What is the proba

Example 3 medium

A spinner has sections: Red (50%), Blue (30%), Green (20%). You spin twice. What is the probability

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