Independent vs. Dependent Events

TipLearning Objectives

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

  • Identify whether events are independent or dependent.
  • Compute probabilities correctly for each type.
  • Understand how replacement affects independence.
  • Recognize dependence in real-world contexts.

Key Ideas

Independent Events

One event does not affect the other.
\[ P(A\ \text{and}\ B) = P(A)P(B) \]

Dependent Events

First event changes the probability of the second.
\[ P(A\ \text{and}\ B) = P(A)\cdot P(B\mid A) \]

Common Problem Types

Testing for Independence

Ask: “Does the first event change the probability of the second?”

Example:
Two coin flips → independent.

Card Draws With and Without Replacement

Replacement → independent
No replacement → dependent

Example:
\(P(\text{ace then ace without replacement}) = (4/52)(3/51)\).

Selecting People or Objects From a Group

Choosing without putting them back typically creates dependence.

Example:
Choosing team captains without replacement.

Situations Where Probabilities Change

Weather, conditional statements, conditional tables.

Example:
\(P(\text{rain tomorrow} \mid \text{rain today})\) differs from \(P(\text{rain tomorrow})\).

Probability Trees for Dependent Events

Tree diagrams show conditional probabilities clearly.

Strategies

  • Ask if the second event’s probability changes after the first.
  • Identify whether replacement occurs.
  • For dependent events, write conditional probability explicitly.
  • Use trees if multiple conditional branches.

Worked Examples

Example 1

Choosing two marbles without replacement from a bag of 5 red, 3 blue.
\(P(\text{red then blue}) = (5/8)(3/7)\).

Example 2

Two independent coin flips:
\(P(\text{heads then tails}) = (1/2)(1/2)=1/4\).

WarningCommon Mistakes
  • Treating dependent events as independent.
  • Forgetting to update totals after removing items.
  • Thinking independence = mutually exclusive (they’re different!).
  • Ignoring conditional probabilities.

Practice Problems

  1. Two coin flips: independent or dependent?
  2. Draw 2 cards with replacement: \(P(\text{two queens})\)?
  3. Draw 2 cards without replacement: \(P(\text{both black})\)?
  4. Choose 2 students from 30 without replacement: dependent?
  1. Independent
  2. \((4/52)(4/52)=1/169\)
  3. \((26/52)(25/51)=25/102\)
  4. Yes, dependent (group size changes).

Summary

  • Independent: one event does not affect the other.
  • Dependent: the first event changes the second.
  • Replacement restores independence.
  • Ask: “Does the first event change the next?”
  • Replacement → independent.
  • Use tree diagrams for conditional probabilities.