Solved Exercises on Compound Events in Grade 8

Master compound events: independent events, dependent events, and probability rules through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Independent Events - Coin Flips
Exercise 1
What is the probability of getting heads on the first flip and tails on the second flip when flipping a fair coin twice?
Definition:

Independent Events: Events where the outcome of one event does not affect the probability of the other event

Independent events method:
  1. Identify that events are independent
  2. Find the probability of each event separately
  3. Multiply the individual probabilities
  4. Simplify the result if necessary
Event A
Heads on first flip
Event B
Tails on second flip
P(A and B)
1/2 × 1/2 = 1/4
Step 1: Identify the events

Event A: Getting heads on the first flip

Event B: Getting tails on the second flip

Step 2: Verify independence

The outcome of the first flip does not affect the second flip

Events A and B are independent

Step 3: Find individual probabilities

P(Heads on first flip) = 1/2

P(Tails on second flip) = 1/2

Step 4: Apply the multiplication rule

P(A and B) = P(A) × P(B)

P(Heads on first AND Tails on second) = 1/2 × 1/2 = 1/4

P(Heads then Tails) = 1/4
Final answer:

The probability of getting heads on the first flip and tails on the second flip is 1/4

Applied rules:

• Independent events: P(A and B) = P(A) × P(B)

• Fair coin: P(Heads) = P(Tails) = 1/2

• Multiplication rule: For independent events

2 Independent Events - Dice Rolls
Exercise 2
What is the probability of rolling a 4 on the first die and a 6 on the second die when rolling two standard dice?
Definition:

Compound Event: An event consisting of two or more simple events occurring together

Die 1
P(rolling 4) = 1/6
Die 2
P(rolling 6) = 1/6
Combined
1/6 × 1/6 = 1/36
Step 1: Identify the events

Event A: Rolling a 4 on the first die

Event B: Rolling a 6 on the second die

Step 2: Verify independence

The outcome of the first die does not affect the second die

Events A and B are independent

Step 3: Find individual probabilities

P(Rolling 4 on first die) = 1/6

P(Rolling 6 on second die) = 1/6

Step 4: Apply the multiplication rule

P(A and B) = P(A) × P(B)

P(4 on first AND 6 on second) = 1/6 × 1/6 = 1/36

P(4 then 6) = 1/36
Final answer:

The probability of rolling a 4 on the first die and a 6 on the second die is 1/36

Applied rules:

• Independent events: P(A and B) = P(A) × P(B)

• Standard die: P(any specific number) = 1/6

• Multiplication rule: For independent events

3 Dependent Events - Drawing Cards
Exercise 3
A standard deck of 52 cards is shuffled. What is the probability of drawing a heart first and then drawing another heart without replacement?
Definition:

Dependent Events: Events where the outcome of one event affects the probability of the other event

First Draw
P(Heart) = 13/52
Second Draw
P(Heart) = 12/51
Combined
13/52 × 12/51 = 1/17
Step 1: Identify the events

Event A: Drawing a heart on the first draw

Event B: Drawing a heart on the second draw

Step 2: Identify dependency

Since we're not replacing the first card, the second draw depends on the first

Events A and B are dependent

Step 3: Find probability of first event

P(Heart on first draw) = 13/52 = 1/4

(There are 13 hearts in a deck of 52 cards)

Step 4: Find probability of second event given first

After drawing one heart, there are 12 hearts left out of 51 cards

P(Heart on second draw | Heart on first) = 12/51

Step 5: Apply the multiplication rule for dependent events

P(A and B) = P(A) × P(B|A)

P(Heart then Heart) = 13/52 × 12/51 = 156/2652 = 1/17

P(Heart then Heart) = 1/17
Final answer:

The probability of drawing a heart first and then drawing another heart without replacement is 1/17

Applied rules:

• Dependent events: P(A and B) = P(A) × P(B|A)

• Without replacement: Total outcomes decrease after each draw

• Conditional probability: P(B|A) means probability of B given A occurred

Compound Events Fundamentals
P(A\ and\ B) = P(A) \times P(B|A)
General Multiplication Rule
Independent
P(A and B) = P(A) × P(B)
Events don't affect each other
Dependent
P(A and B) = P(A) × P(B|A)
Events affect each other
Mutually Exclusive
P(A and B) = 0
Cannot occur together
Key definitions:

Compound Event: An event that consists of two or more simple events occurring together

Independent Events: Events where the outcome of one event does not affect the probability of another

Dependent Events: Events where the outcome of one event affects the probability of another

Conditional Probability: The probability of an event occurring given that another event has occurred

Mutually Exclusive Events: Events that cannot occur at the same time

Replacement: Returning an item to the original set after selection

Without Replacement: Not returning an item to the original set after selection

Compound Events Process:
  1. Identify events: Determine what events are occurring
  2. Check independence: Determine if events are independent or dependent
  3. Select rule: Use appropriate multiplication rule
  4. Calculate probabilities: Find individual probabilities
  5. Apply rule: Multiply according to independence
  6. Simplify: Reduce fraction if possible
Tip 1: Look for keywords like "without replacement" to identify dependent events.
Tip 2: With replacement usually means independent events.
Tip 3: Always consider how the first event affects the second event.
Tip 4: Tree diagrams can help visualize compound events.
Common errors: Treating dependent events as independent, forgetting to adjust for removal.
Exam preparation: Practice distinguishing between independent and dependent events, apply correct rules.
Solution: Exercises 4 to 5
4 Dependent Events - Marbles
Exercise 4
A bag contains 5 red marbles and 3 blue marbles. Two marbles are drawn without replacement. What is the probability of drawing two red marbles?
Definition:

Without Replacement: The first item is not returned to the set before the second selection

First Draw
P(Red) = 5/8
Second Draw
P(Red) = 4/7
Combined
5/8 × 4/7 = 5/14
Step 1: Identify the events

Event A: Drawing a red marble on the first draw

Event B: Drawing a red marble on the second draw

Step 2: Note the dependency

Since we're drawing without replacement, the second draw depends on the first

Events A and B are dependent

Step 3: Find probability of first event

P(Red on first draw) = 5/8

(5 red marbles out of 8 total marbles)

Step 4: Find probability of second event given first

After drawing one red marble: 4 red marbles remain out of 7 total marbles

P(Red on second draw | Red on first) = 4/7

Step 5: Apply the multiplication rule for dependent events

P(A and B) = P(A) × P(B|A)

P(Red then Red) = 5/8 × 4/7 = 20/56 = 5/14

P(Red then Red) = 5/14
Final answer:

The probability of drawing two red marbles without replacement is 5/14

Applied rules:

• Dependent events: P(A and B) = P(A) × P(B|A)

• Without replacement: Total marbles decrease after first draw

• Adjustment: Both numerator and denominator change

5 Compound Events - With Replacement
Exercise 5
Using the same bag of 5 red and 3 blue marbles, what is the probability of drawing a red marble first and a blue marble second WITH replacement?
Definition:

With Replacement: The first item is returned to the set before the second selection

First Draw
P(Red) = 5/8
Second Draw
P(Blue) = 3/8
Combined
5/8 × 3/8 = 15/64
Step 1: Identify the events

Event A: Drawing a red marble on the first draw

Event B: Drawing a blue marble on the second draw

Step 2: Note the independence

Since we're drawing WITH replacement, the second draw is independent of the first

Events A and B are independent

Step 3: Find probability of first event

P(Red on first draw) = 5/8

(5 red marbles out of 8 total marbles)

Step 4: Find probability of second event

Since the first marble is replaced, the bag is restored to its original state

P(Blue on second draw) = 3/8

Step 5: Apply the multiplication rule for independent events

P(A and B) = P(A) × P(B)

P(Red then Blue with replacement) = 5/8 × 3/8 = 15/64

P(Red then Blue) = 15/64
Final answer:

The probability of drawing a red marble first and a blue marble second with replacement is 15/64

Applied rules:

• Independent events: P(A and B) = P(A) × P(B)

• With replacement: Probabilities remain constant

• Independence: First draw doesn't affect second draw

Compound Events Analysis Summary
P(A\ and\ B) = P(A) \times P(B\mid A)
General Multiplication Rule
Key definitions:

Compound Event: An event consisting of two or more simple events occurring together

Independent Events: Events where P(B|A) = P(B), meaning A does not affect B

Dependent Events: Events where P(B|A) ≠ P(B), meaning A affects B

Conditional Probability: P(B|A) represents the probability of B given that A has occurred

With Replacement: The item is returned after selection, keeping probabilities constant

Without Replacement: The item is not returned, changing probabilities for subsequent selections

Mutually Exclusive: Events that cannot occur simultaneously, P(A and B) = 0

Complete Compound Events Analysis:
  1. Event identification: Clearly define what events are occurring
  2. Independence check: Determine if events are independent or dependent
  3. Rule selection: Choose appropriate multiplication rule
  4. Probability calculation: Find individual probabilities
  5. Rule application: Multiply according to the independence relationship
  6. Verification: Check that the answer makes sense
Tip 1: Look for "with replacement" vs "without replacement" to determine independence.
Tip 2: For dependent events, adjust the sample space after each selection.
Tip 3: The probability of compound events is always less than or equal to the probability of individual events.
Tip 4: Tree diagrams help visualize multiple-step probability problems.
Applications: Used in games, insurance, finance, and risk assessment.
Limitations: Assumes perfect randomness; real-world conditions may vary.
Essential Formulas:

• Independent events: P(A and B) = P(A) × P(B)

• Dependent events: P(A and B) = P(A) × P(B|A)

• General rule: P(A and B) = P(A) × P(B|A)

• Mutually exclusive: P(A and B) = 0

Questions & Answers

Question: How can I tell if events are independent or dependent? What are some clues?

Answer: Look for these clues:

  • Independent: Keywords like "with replacement", "returned", "separate", or when events occur in isolation
  • Dependent: Keywords like "without replacement", "not returned", "drawn from same set"

Ask yourself: "Does the outcome of the first event change the conditions for the second event?" If yes, they're dependent. If no, they're independent.

Examples:
Independent: Rolling a die twice (first roll doesn't affect second)
Dependent: Drawing cards without replacement (first card affects remaining cards)

Question: Why is the probability of compound events usually smaller than individual event probabilities?

Answer: When multiplying probabilities (which are typically fractions between 0 and 1), the result is smaller than the original values:

For example: P(A) = 1/2 and P(B) = 1/3

P(A and B) = 1/2 × 1/3 = 1/6

Since 1/6 < 1/2 and 1/6 < 1/3, the compound probability is smaller.

This makes sense logically: it's harder for two specific events to both happen than for just one of them to happen. The more conditions you add, the less likely the overall outcome becomes.

Question: What if I have more than two events? How do I calculate compound probability for three or more events?

Answer: Extend the multiplication rule for multiple events:

  • Independent events: P(A and B and C) = P(A) × P(B) × P(C)
  • Dependent events: P(A and B and C) = P(A) × P(B|A) × P(C|A and B)

For example, if rolling three dice independently to get three 6's:

P(6 and 6 and 6) = 1/6 × 1/6 × 1/6 = 1/216

For dependent events, adjust the probability at each step based on what has already occurred.

Question: What's the difference between "mutually exclusive" and "independent" events? They seem similar.

Answer: These are completely different concepts:

  • Mutually exclusive: Events that cannot happen at the same time
  • Independent: Events where one doesn't affect the probability of the other

Mutually exclusive events are actually dependent! If A and B are mutually exclusive, then P(B|A) = 0, which is not equal to P(B).

Example: Rolling a die
Mutually exclusive: Rolling a 3 AND rolling a 5 (impossible)
Independent: Rolling a 3 on first die AND rolling a 5 on second die

Question: How do tree diagrams help with compound events? When should I use them?

Answer: Tree diagrams are excellent for visualizing compound events:

Benefits:
- Show all possible outcomes clearly
- Help track how probabilities change at each stage
- Make it easy to see dependent vs independent events

Use tree diagrams when:
- There are multiple stages to the experiment
- You need to visualize all possible outcomes
- Working with dependent events
- The problem involves sequential decisions

They're especially helpful for "without replacement" problems and multi-step probability scenarios.