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Teacher Guide: Dependent Events

Learn how one event affects the probability of another when outcomes are connected.

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Printable worksheet

All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Compound Events. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives
  • Distinguish between dependent and independent events
  • Calculate probabilities of dependent events using conditional probability
  • Apply the multiplication rule for dependent events
  • Solve problems involving selection without replacement
  • Recognize dependent events in real-world contexts
Prerequisites
  • • Understanding of basic probability concepts
  • • Ability to multiply fractions
  • • Familiarity with sample space and outcomes
  • • Knowledge of independent events (helpful but not required)
Discussion Starters
  • 1. If you draw a card from a deck and don't put it back, how does this change the next draw?
  • 2. Why do card counters in casinos track which cards have been played?
  • 3. Can you think of a real-life situation where events are clearly dependent?
  • 4. How would the probability change if we replaced the card after each draw?
Common Misconceptions

The probability stays the same for each draw

Remediation: Use physical demonstrations with cards or marbles. Show that after removing a heart, only 12 hearts remain out of 51 cards.

Only the denominator changes, not the numerator

Remediation: Draw two hearts scenario: First draw takes 1 heart AND 1 card from total. Both 13 hearts and 52 cards decrease.

All compound events are dependent

Remediation: Compare: Drawing without replacement (dependent) vs rolling two dice (independent). First die result doesn't change second die options.

Differentiation Ideas

For Struggling Students:

  • • Use physical manipulatives (actual cards, colored counters)
  • • Start with smaller sets (5-10 items) before moving to 52-card deck
  • • Create visual tree diagrams showing all possibilities
  • • Practice with two-event problems before three-event problems

For On-Level Students:

  • • Calculate probabilities with standard deck problems
  • • Compare dependent vs independent event calculations
  • • Work with word problems involving committees and selections
  • • Extend to three consecutive dependent events

For Advanced Students:

  • • Explore conditional probability notation P(B∣A)P(B|A) formally
  • • Investigate Bayes' theorem introduction
  • • Analyze complex multi-stage problems
  • • Calculate expected values with dependent probabilities
Standards Alignment
  • S-CP.A.3 (CCSS.MATH.CONTENT.HSS.CP.A.3)

    Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B

  • S-CP.A.5 (CCSS.MATH.CONTENT.HSS.CP.A.5)

    Recognize and explain the concepts of conditional probability and independence in everyday language

  • S-CP.B.8 (CCSS.MATH.CONTENT.HSS.CP.B.8)

    Apply the general Multiplication Rule in a uniform probability model

Lesson Resources
  • visualCard Deck Simulator

    Draw cards without replacement and see probabilities change

  • activityMarble Bag Experiment

    Physical demonstration with colored marbles

  • worksheetDependent vs Independent

    Classify events and calculate probabilities

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

Dependent events are events where the outcome of one event affects the probability of another event.
When events are dependent:
  • The first event changes the sample space for the second event
  • We use the formula: P(A and B)=P(A)×P(B∣A)P(A \text{ and } B) = P(A) \times P(B | A)
The notation P(B∣A)P(B | A) means "the probability of B given that A has occurred."
Key difference from independent events:
  • Independent: P(A and B)=P(A)×P(B)P(A \text{ and } B) = P(A) \times P(B)
  • Dependent: P(A and B)=P(A)×P(B∣A)P(A \text{ and } B) = P(A) \times P(B | A)

Worked Examples

A standard deck has 52 cards. You draw 2 cards without replacement. What is the probability that both cards are hearts?
1

Find P(first heart)

There are 13 hearts out of 52 cards → P(H1)=1352=14P(H_1) = \frac{13}{52} = \frac{1}{4}
2

Find P(second heart | first was heart)

After drawing 1 heart: 12 hearts remain, 51 cards total → P(H2∣H1)=1251P(H_2 | H_1) = \frac{12}{51}
3

Multiply the probabilities

P(H1 and H2)=1352×1251P(H_1 \text{ and } H_2) = \frac{13}{52} \times \frac{12}{51} → =1562652= \frac{156}{2652}
4

Simplify

1562652=117\frac{156}{2652} = \frac{1}{17} → ≈0.059\approx 0.059 or about 5.9%5.9\%

Common Mistakes

Using the same denominator for both events

Why it's wrong: When drawing without replacement, the total number of items decreases. Using 52 for both card draws ignores this change.

Correct: After the first draw, reduce the denominator. Second draw from 51 cards, third draw from 50 cards, etc.

Forgetting to adjust the numerator

Why it's wrong: If you draw a heart first, there are only 12 hearts left, not 13. The first event changes what remains.

Correct: If drawing the same category, reduce both numerator AND denominator. If different category, reduce only the denominator.

Treating dependent events as independent

Why it's wrong: Using P(A)×P(B)P(A) \times P(B) instead of P(A)×P(B∣A)P(A) \times P(B|A) gives wrong answers for dependent events.

Correct: Always ask: Does the first event change what's available for the second? If yes, events are dependent.

Why It Matters

Dependent events appear constantly in real-world scenarios:
  • Card games: Drawing cards without replacement changes the deck composition
  • Quality control: Testing items from a batch affects what remains
  • Weather forecasting: Today's weather affects tomorrow's probability
  • Medical testing: Previous test results influence subsequent diagnoses
  • Sports: A player's fatigue affects performance probability in later games
Understanding dependent events prevents costly mistakes in probability calculations!

Real World Applications

Quality Control Testing

Factories test product samples. Each tested item is removed from the batch, making subsequent selections dependent.

Example:

A box has 100 lightbulbs, 5 defective. Testing 3 without replacement: P(all good)=95100×9499×9398P(\text{all good}) = \frac{95}{100} \times \frac{94}{99} \times \frac{93}{98}

1Try It Yourself

A batch of 20 phones has 2 defective. You test 2 phones without replacement.

What is the probability that both phones tested are NOT defective?

Step 1: Write the mathematical expression

Calculate: 1820×??\frac{18}{20} \times \frac{?}{?}

Card Game Strategies

Professional card players track which cards have been played to calculate remaining probabilities.

Example:

In blackjack, if 3 aces have been dealt from a single deck, the probability of the next card being an ace is 149\frac{1}{49}, not 452\frac{4}{52}.

2Try It Yourself

From a standard deck, 10 cards have been dealt. You saw 3 spades among them.

What is the probability the next card is a spade?

Step 1: Write the mathematical expression

Remaining spades / Remaining cards = ?

Selecting Committee Members

When choosing people for committees, each selection reduces the pool for subsequent choices.

Example:

From 8 men and 6 women, selecting 2 people: P(both women)=614×513P(\text{both women}) = \frac{6}{14} \times \frac{5}{13}

3Try It Yourself

A club has 10 members: 4 seniors and 6 juniors. Two officers are chosen at random.

What is the probability that both officers are seniors?

Step 1: Write the mathematical expression

Calculate the dependent probability:

Key Takeaways

  • 1Dependent events are when one event affects the probability of another
  • 2Use the formula: P(A and B)=P(A)×P(B∣A)P(A \text{ and } B) = P(A) \times P(B | A)
  • 3Without replacement means the sample space shrinks after each event
  • 4Always adjust both the numerator (items of interest) and denominator (total items) after each event
  • 5Ask yourself: Does the first outcome change what's possible for the second? If yes, events are dependent

Frequently Asked Questions

How do I know if events are dependent or independent?

Ask: Does the first event change what's possible for the second? Drawing cards without replacement is dependent (deck shrinks). Rolling dice twice is independent (first roll doesn't affect second).

What does 'without replacement' mean?

It means you don't put the item back after selecting it. This changes the probabilities for subsequent selections because both the total count and possibly the count of desired items decrease.

Can events be dependent even with replacement?

Yes! If the outcome of one event gives you information that changes the probability of another, they're dependent. For example, knowing it rained today affects the probability it rains tomorrow.

Glossary

Dependent events
Events where the outcome of one affects the probability of another
Independent events
Events where the outcome of one has no effect on the probability of another
Conditional probability
The probability of an event given that another event has occurred, written as P(B∣A)P(B|A)
Without replacement
Selecting items without returning them to the original set
Sample space
The set of all possible outcomes

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