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Probability of Marbles in a Jar

Aug 15, 2025

Overview

This lecture covers solving probability problems involving selecting colored marbles from a jar, focusing on "and" vs. "or" probability rules, and both with- and without-replacement scenarios.

Jar Composition & Total Calculation

  • The jar contains 7 red, 6 green, 5 blue, and 2 yellow marbles.
  • Total number of marbles: 7 + 6 + 5 + 2 = 20.

Calculating Basic Probabilities

  • Probability of green: 6/20 = 3/10 = 0.3 (30%).
  • Probability of blue: 5/20 = 1/4 = 0.25 (25%).
  • Probability of not blue: 1 - 0.25 = 0.75 (75%).
  • Probability of not green: marbles not green = 7 + 5 + 2 = 14; so 14/20 = 7/10 = 0.7 (70%).

"Or" Probabilities

  • Probability of green or yellow: (6/20) + (2/20) = 8/20 = 2/5 = 0.4 (40%).

"And" Probabilities with Replacement

  • Probability of red then blue (with replacement): (7/20) × (5/20) = 35/400 = 7/80 ≈ 0.0875 (8.75%).

"And" Probabilities without Replacement

  • Probability of red then blue (without replacement): (7/20) × (5/19) = 35/380 = 7/76 ≈ 0.092 (9.2%).
  • Probability of red and blue in any order (without replacement): 2 × (7/76) = 14/76 = 7/38 ≈ 0.184 (18.4%).
  • Probability of not red and blue in any order: 1 - 0.184 = 0.816 (81.6%).

Complex Combined Scenarios

  • Probability of (red then green) or (blue then yellow) with replacement:
    • (7/20 × 6/20) + (5/20 × 2/20) = (42/400) + (10/400) = 52/400 = 13/100 = 0.13 (13%).

Key Terms & Definitions

  • Probability — Likelihood of an event, calculated as favorable outcomes divided by total possible outcomes.
  • With Replacement — After selecting a marble, it is returned to the jar, keeping the total number unchanged.
  • Without Replacement — Marble is not returned after selection, so the total number decreases.
  • "And" Probability — Multiply probabilities for dependent/independent events occurring together.
  • "Or" Probability — Add probabilities for either of two events occurring.

Action Items / Next Steps

  • Practice identifying "and" vs. "or" scenarios in different probability questions.
  • Try additional probability problems using both with- and without-replacement methods.

Full transcript