The Card Sequence Problem
Problem Statement
You have three cards numbered 1, 2, and 3. You shuffle them and place them face down in a row.
Question: What is the probability that at least one card is in its correct position (i.e., card number matches its position)?
Hints:
- Start by listing all possible arrangements (permutations) of three cards
- For each arrangement, check if any card is in its correct position
- Remember: Total probability = Favorable outcomes / Total outcomes
Step-by-Step Solution
1. List All Possible Arrangements
With 3 cards, we have 3! = 6 possible arrangements:
2. Count Favorable Outcomes
Let’s analyze each arrangement:
- 1,2,3 → All three cards match (1 case)
- 1,3,2 → Only card 1 matches (1 case)
- 2,1,3 → Only card 3 matches (1 case)
- 2,3,1 → No matches (0 case)
- 3,1,2 → No matches (0 case)
- 3,2,1 → No matches (0 case)
3. Calculate Probability
Total favorable outcomes = 3 arrangements have at least one match
Total possible outcomes = 6 arrangements
Therefore, probability = 3/6 = 1/2 = 0.5 = 50%
Interesting Note
This is related to the concept of “derangements” in probability – arrangements where no element appears in its original position.
Verify Understanding
You can check your answer by:
- Drawing a tree diagram
- Using the complement: P(at least one match) = 1 – P(no matches)
- Simulating the problem with actual cards
Weekly Problem: The Card Sequence Problem
