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Independent vs Dependent

Overview

Does the past matter?

What is it?

Independent: One event does not affect the next. Dependent: The first event changes the odds for the second.

History

Critical in the Monty Hall Problem, which confuses even Ph.D. mathematicians.

Key Idea

Replacement = Independent. No Replacement = Dependent.

Practice This Topic

Concept Guide

Plain English: If you flip a coin, the coin doesn't care what happened last time (Independent). If you eat a chocolate from a box, that chocolate is gone, so the odds of picking a caramel next time have changed (Dependent).

Real-world example: Card Counting in Blackjack. Since cards aren't put back in the deck, the odds change with every card dealt.

How to do it

  1. Ask: 'Did the total number of items change?'
  2. If Yes (ate candy, kept card): Dependent. Subtract 1 from total for next fraction.
  3. If No (coin flip, dice roll): Independent. Keep fractions the same.
  4. Multiply the fractions.
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Common Pitfall

Treating Dependent events as Independent. If you pick two cards, you can't just say 1/52 * 1/52. The second pick is 1/51.

Word Problem
"You have 2 Aces in a 52 card deck. You draw one and keep it. What are odds of drawing the second Ace?"
Reasoning: First Draw: 2/52. Second Draw (Dependent): Only 1 Ace left, only 51 cards left. 1/51.

Practice Examples

Dependent
P(A and B) = P(A) * P(B|A)
Prob of B GIVEN A happened.