Teacher Guide: Dependent Events
Learn how one event affects the probability of another when outcomes are connected.
Use this lesson with your class
Free, no student accounts needed.
Share with students
Students open the lesson and practise with instant feedback.
Class quiz
10 questions on Compound Events. Students join with a name, you see everyone's score.
For Teachers
- 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
- • Understanding of basic probability concepts
- • Ability to multiply fractions
- • Familiarity with sample space and outcomes
- • Knowledge of independent events (helpful but not required)
- 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?
The probability stays the same for each draw
Only the denominator changes, not the numerator
All compound events are dependent
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 formally
- • Investigate Bayes' theorem introduction
- • Analyze complex multi-stage problems
- • Calculate expected values with dependent probabilities
- 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
- 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.
Lesson Content
Everything students see: definition, examples, common mistakes, applications. Tap to open.
Definition
- The first event changes the sample space for the second event
- We use the formula:
- Independent:
- Dependent:
Worked Examples
A standard deck has 52 cards. You draw 2 cards without replacement. What is the probability that both cards are hearts?
Find P(first heart)
There are 13 hearts out of 52 cards →
Find P(second heart | first was heart)
After drawing 1 heart: 12 hearts remain, 51 cards total →
Multiply the probabilities
→
Simplify
→ or about
Answer: The probability of drawing 2 hearts in a row is or approximately
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 instead of 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
- 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
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:
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:
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 , not .
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:
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:
- 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?
What does 'without replacement' mean?
Can events be dependent even with replacement?
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
- Without replacement
- Selecting items without returning them to the original set
- Sample space
- The set of all possible outcomes