Probability Word Problems

Introduction to Probability Word Problems

• Probability covered in algebra/pre-calculus courses.
• Focus on solving basic word problems.
• Key concept: understanding events as independent or dependent.

Problem 1: Independent Events

• Scenario: Jason flips a coin and rolls a six-sided die.
• Question: Probability of coin landing heads and die showing an even number.
• Key Concept: "And" implies multiplication for independent events.
  • Probability Formula: P(A and B) = P(A) * P(B)
  • Flipping a coin (Heads) = 1/2
  • Die rolling an even number (2, 4, 6) = 3/6 = 1/2
  • Result: 1/2 * 1/2 = 1/4 or 25%.

Problem 2: Dependent Events

• Scenario: Allison selects three students from her class.
• Question: Probability first student is a boy, second and third are girls.
• Key Concept: Adjust probabilities since events are dependent.
  • Formula: P(A, then B, then C) = P(A) * P(B|A) * P(C|A and B)
  • Initial probabilities: Boy = 5/11, Girl = 6/10, Girl = 5/9 (adjusted each time)
  • Result: 5/11 * 6/10 * 5/9 = 5/33.

Problem 3: "Or" Events

• Scenario: Katie's selection of a coin.
• Question: Probability it is a dime or from the US.
• Key Concept: "Or" implies addition but subtract overlap (non-mutually exclusive).
  • Formula: P(A or B) = P(A) + P(B) - P(A and B)
  • Dime = 7/13, US coin = 7/13, Dime and US = 6/13
  • Result: 8/13.

Problem 4: Mutually Exclusive Events

• Scenario: Beth selects fruit.
• Question: Probability it is an apple or a peach.
• Key Concept: If mutually exclusive, no overlap subtraction needed.
  • Apples = 4/13, Peaches = 4/13
  • Result: 8/13.

Problem 5: Binomial Probability

• Scenario: Desk assignment in a classroom over 5 days.
• Question: Probability of sitting in the front row exactly 3 times.
• Key Concept: Use binomial probability formula for repeated independent trials.
  • Formula: ( nC_r \times p^r \times (1-p)^{n-r} )
  • Probability of front row = 1/4
  • Calculation: 5 choose 3, probability raised to r and n-r powers.
  • Result: 45/512.