Lecture Notes: Probability Questions

Introduction

• Focus on five hardest probability questions.
• Involve algebra, probability trees, different methods.
• Questions in no particular order of difficulty.

Question 1

• Scenario: Bag with r red and g green counters.
  • Probability green = 3/7 initially.
  • Two red and three green added; new probability green = 6/13.
• Objective: Find original r and g.
• Method:
  • Use ratios: Green to Red = 3:4, use 3x and 4x.
  • After adding counters, green becomes 3x + 3, red becomes 4x + 2.
  • Total counters: 7x + 5.
  • Equation from probability: (3x + 3) / (7x + 5) = 6/13.
  • Solve for x: Cross-multiply, simplify to find x = 3.
  • Original count: Green 3x = 9, Red 4x = 12.

Question 2

• Scenario: 10 pens, x red, rest blue.
  • Find expression for probability of one pen of each color.
• Method:
  • Ratio: Red to Blue = x to 10 - x.
  • Use probability tree to outline possible outcomes.
  • Calculate probabilities for each path: (x/10) * ((10-x)/9) and ((10-x)/10) * (x/9).
  • Add probabilities, simplify: 10x - x² / 45.

Question 3

• Scenario: More than 12 pens, 3 more blue than green.
  • Probability of two same color pens = 27/55.
• Method:
  • Ratios: Blue = x+3, Green = x, total = 2x+3.
  • Set up probability tree, focus on same color paths.
  • Equations for probabilities, cross-multiply.
  • Solve quadratic: 2x² - 50x + 168 = 0.
  • Factorize or use quadratic formula to find x = 21.

Question 4

• Scenario: y black socks, 5 white socks.
  • Probability of one white, one black = 6/11.
  • Show quadratic: 3y² - 28y + 60 = 0.
• Method:
  • Setup probability expressions, solve using given probability.
  • Factorize or use quadratic formula to solve quadratic.
  • Find probability for two black socks using calculated y.

Question 5

• Scenario: 7 red counters, rest white.
  • Probability first white, then red = 21/80.
• Method:
  • Expression for white and red, probability tree.
  • Cross-multiply, create quadratic.
  • Solve using quadratic formula: x = 9 (number of white counters).

Conclusion

• These questions demonstrate complex probability involving algebra.
• Practice with similar problems recommended.
• Videos available for more detailed walkthroughs on specific topics.