Probability of Matching Socks

Overview

• Discussing a scenario involving choosing pairs of socks at random from a drawer.
• Objective: Find the probability of wearing a matching pair of socks.

Key Concepts

• Probability: A fraction of favorable outcomes over the sample space.
• Favorable outcomes: Outcomes that meet the desired criteria (matching pairs).
• Sample space: Total possible outcomes (total socks available).

Scenario Breakdown

1. Initial Setup:

   • 10 socks in total, consisting of 5 pairs of different colors.
   • Each morning, one randomly picks 2 socks.

2. First Morning Probability Calculation:

   • Step 1: Choose the first sock (doesn't matter which one).
   • Step 2: Choose the second sock (aim for a match).
   • Sample space changes:
     • After choosing the first sock, 9 socks remain.
   • Favorable outcome: Only 1 sock matches the first chosen sock.
   • Probability formula:
     • Probability (P) = Favorable outcomes / Sample space = 1/9.

Questions on Follow-Up Days

• Next Day Probability:
  • Probability remains 1/9 due to the same reasoning of selection.
  • Key consideration: The process remains independent of previous selections.

Discussion Points

• Replacement vs. Non-Replacement:
  • The probability does not change as the scenario describes it without replacement.
  • Importance of re-evaluating the scenario based on the wording of the question.
  • Clarification needed on dynamic probability and its implications over multiple days.

Conclusion

• Probability of wearing a matching pair of socks remains consistent at 1/9, regardless of the day.
• The exercise demonstrates the importance of understanding probability fundamentals and carefully interpreting problems.