Compound Probability

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Compute probabilities of “AND” and “OR” events.
  • Distinguish between mutually exclusive and non-exclusive events.
  • Use complements for multi-step events.
  • Apply rules to dice, cards, spinners, and tables.

Key Ideas

“AND” = multiply

\[ P(A\ \text{and}\ B) = P(A) \cdot P(B) \]

“OR” = add

If mutually exclusive: \[ P(A\ \text{or}\ B) = P(A) + P(B) \]

If NOT mutually exclusive: \[ P(A\ \text{or}\ B) = P(A) + P(B) - P(A\cap B) \]

Common Problem Types

“AND” With Independent Events

Multiply probabilities.

Example:
Flip two coins: \(P(\text{HH}) = 1/2 \cdot 1/2 = 1/4\).

“AND” With Dependent Events

Probability changes after first event.

Example:
Drawing two cards without replacement.

“OR” Mutually Exclusive Events

Add probabilities.

Example:
Roll a die: \(P(1 \text{ or } 6) = 1/6 + 1/6 = 1/3\).

“OR” Overlapping Events

Subtract double-counted overlap.

Example:
Drawing a face card OR a heart.
Need to subtract face-card hearts.

Using Complements for Multi-Step Problems

Often easier than enumerating outcomes.

Example:
At least one head in 3 flips =
\(1 - P(\text{no heads}) = 1 - (1/2)^3 = 7/8\).

Strategies

  • Identify if events overlap.
  • Use Venn diagrams when unsure.
  • Decide whether replacement affects probabilities.
  • For “at least one,” start with complement.

Worked Examples

Example 1

Two dice: probability they sum to 7?

Outcomes: (1,6), (2,5), (3,4), etc. → 6/36 = 1/6.

Example 2

Drawing 2 cards without replacement, probability both are aces?

\[ \frac{4}{52} \cdot \frac{3}{51} = \frac{12}{2652} = \frac{1}{221} \]

WarningCommon Mistakes
  • Adding when you should multiply.
  • Forgetting to subtract the overlap in “OR” problems.
  • Ignoring dependence when there is no replacement.
  • Assuming events are mutually exclusive when they are not.

Practice Problems

  1. Flip two coins. \(P(\text{two heads})\)?
  2. Roll a die. \(P(\text{even or prime})\)?
  3. Draw two cards without replacement: \(P(\text{both red})\)?
  4. Find \(P(\text{at least one head in 2 flips})\).
  1. \(1/2 \cdot 1/2 = 1/4\)
  2. Even = {2,4,6}, prime = {2,3,5}, overlap = 2 → \(3/6 + 3/6 - 1/6 = 5/6\)
  3. Red cards = 26 → \((26/52)(25/51)=25/102\)
  4. \(1 - P(\text{no heads}) = 1 - (1/2)^2 = 3/4\)

Summary

  • AND = multiply. OR = add (minus overlap).
  • Use complements for efficiency.
  • Check independence/dependence.
  • Determine: independent? mutually exclusive?
  • Draw Venn diagrams for overlapping events.
  • Use complement for “at least one.”