Solved Exercises on Compound Events in Grade 7

Master compound probability: independent events, dependent events, tree diagrams, and probability rules through these 10 detailed exercises with visual learning tools.

Solution: Exercises 1 to 3
1 Independent Events
Exercise 1
What is the probability of rolling a 3 on a standard die and flipping heads on a coin?
Definition:

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

Method:

For independent events A and B: P(A and B) = P(A) × P(B)

Step 1: Find probability of rolling a 3

P(rolling 3) = 1/6

Step 2: Find probability of flipping heads

P(heads) = 1/2

Step 3: Multiply probabilities

P(3 and heads) = (1/6) × (1/2) = 1/12

Roll 3: 1/6
×
Flip heads: 1/2
= 1/12
Probability: 1/12
Final answer:

The probability of rolling a 3 and flipping heads is 1/12.

Applied rules:

• Independence: Outcomes don't affect each other

• Multiplication rule: P(A and B) = P(A) × P(B)

• Individual probabilities: Calculate each separately

2 Drawing Cards with Replacement
Exercise 2
A card is drawn from a deck of 52 cards, replaced, and then a second card is drawn. What is the probability that both cards are hearts?
Definition:

With replacement: The first card is returned to the deck before drawing the second card, keeping probabilities constant.

Step 1: Find probability of drawing a heart

P(heart) = 13/52 = 1/4

Step 2: Since card is replaced, probability stays the same

P(second heart) = 1/4

Step 3: Multiply probabilities

P(both hearts) = (1/4) × (1/4) = 1/16

First draw: 1/4
×
Second draw: 1/4
= 1/16
Probability: 1/16
Final answer:

The probability of drawing two hearts with replacement is 1/16.

Applied rules:

• With replacement: Probabilities remain constant

• Independent events: First draw doesn't affect second draw

• Multiplication: P(A and B) = P(A) × P(B)

3 Multiple Spinners
Exercise 3
Two spinners are spun. Spinner 1 has 4 equal sections numbered 1-4. Spinner 2 has 3 equal sections numbered 1-3. What is the probability of spinning a 2 on the first spinner and a 3 on the second spinner?
Definition:

Independent spinners: Each spinner operates independently, so the outcome of one doesn't affect the other.

Step 1: Find probability of spinning 2 on first spinner

P(spinning 2 on spinner 1) = 1/4

Step 2: Find probability of spinning 3 on second spinner

P(spinning 3 on spinner 2) = 1/3

Step 3: Multiply probabilities

P(2 and 3) = (1/4) × (1/3) = 1/12

Spinner 1: 1/4
×
Spinner 2: 1/3
= 1/12
Probability: 1/12
Final answer:

The probability of spinning a 2 on the first spinner and a 3 on the second spinner is 1/12.

Applied rules:

• Independent events: Each spinner is independent

• Equal sections: Each outcome is equally likely

• Multiplication rule: P(A and B) = P(A) × P(B)

Solution: Exercises 4 to 6
4 Dependent Events
Exercise 4
A bag contains 5 red marbles and 3 blue marbles. Two marbles are drawn without replacement. What is the probability that both marbles are red?
Definition:

Without replacement: The first marble is not returned to the bag, so the second draw has different probabilities.

Step 1: Find probability of first red marble

P(first red) = 5/8

Step 2: After drawing one red, find probability of second red

P(second red | first was red) = 4/7 (4 red left out of 7 total)

Step 3: Multiply probabilities

P(both red) = (5/8) × (4/7) = 20/56 = 5/14

First draw: 5/8
×
Second draw: 4/7
= 5/14
Probability: 5/14
Final answer:

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

Applied rules:

• Without replacement: Probabilities change after each draw

• Conditional probability: Second event depends on first

• Multiplication: P(A and B) = P(A) × P(B|A)

5 Conditional Probability
Exercise 5
A class has 12 boys and 18 girls. If two students are selected randomly without replacement, what is the probability that both are girls?
Definition:

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

Step 1: Find probability of selecting first girl

P(first girl) = 18/30 = 3/5

Step 2: After selecting one girl, find probability of second girl

P(second girl | first was girl) = 17/29

Step 3: Multiply probabilities

P(both girls) = (3/5) × (17/29) = 51/145

First selection: 18/30
×
Second selection: 17/29
= 51/145
Probability: 51/145
Final answer:

The probability of selecting two girls without replacement is 51/145.

Applied rules:

• Without replacement: Total decreases after each selection

• Conditional probability: Second probability depends on first outcome

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

6 Tree Diagram Problem
Exercise 6
A coin is flipped twice. Use a tree diagram to find the probability of getting exactly one head.
Definition:

Tree diagram: A visual representation showing all possible outcomes of sequential events.

Step 1: Draw tree diagram showing all outcomes

First flip: H or T, Second flip: H or T for each first outcome

Step 2: Identify outcomes with exactly one head

Outcomes: HT and TH

Step 3: Calculate probability of each outcome

P(HT) = (1/2) × (1/2) = 1/4, P(TH) = (1/2) × (1/2) = 1/4

Step 4: Add probabilities of favorable outcomes

P(exactly one head) = 1/4 + 1/4 = 1/2

Start
H (1/2)
H (1/2)
HH
T (1/2)
HT
T (1/2)
H (1/2)
TH
T (1/2)
TT
HT: 1/4
+
TH: 1/4
= 1/2
Probability: 1/2
Final answer:

The probability of getting exactly one head in two flips is 1/2.

Applied rules:

• Tree diagram: Visualize all possible outcomes

• Independent events: Each flip is independent

• Addition rule: Add probabilities of mutually exclusive outcomes

Compound Probability Visual Guide
P(A and B) = P(A) × P(B), P(A or B) = P(A) + P(B) - P(A and B)
Key Probability Formulas
Independent Events
P(A and B) = P(A) × P(B)
Dependent Events
P(A and B) = P(A) × P(B|A)
Addition Rule
P(A or B) = P(A) + P(B) - P(A and B)
Conditional Probability
P(B|A) = P(A and B) / P(A)
Probability Rules Process:
Step 1: Determine if events are independent or dependent
Step 2: Identify if you need "and" or "or" probability
Step 3: Apply the appropriate formula
Step 4: Calculate individual probabilities
Step 5: Perform the required operation
Step 6: Simplify the result if possible
Tip 1: Independent events don't affect each other's probability.
Tip 2: Dependent events change the sample space for subsequent events.
Tip 3: Use tree diagrams to visualize all possible outcomes.
Common errors: Confusing independent/dependent events, using wrong formulas, miscalculating conditional probabilities.
Success strategies: Identify event type first, use systematic approaches, verify reasonableness of answers.
Essential concepts:

• Probability: Number between 0 and 1

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

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

• Mutually exclusive: P(A and B) = 0

Solution: Exercises 7 to 10
7 Real-World Compound Probability
Exercise 7
In a shipment of 100 light bulbs, 5 are defective. If two bulbs are selected randomly without replacement, what is the probability that both are defective?
Definition:

Real-world probability: Applying probability concepts to practical situations like quality control.

Step 1: Find probability of first bulb being defective

P(first defective) = 5/100 = 1/20

Step 2: After removing one defective bulb, find probability of second defective

P(second defective | first was defective) = 4/99

Step 3: Multiply probabilities

P(both defective) = (1/20) × (4/99) = 4/1980 = 1/495

First bulb: 5/100
×
Second bulb: 4/99
= 1/495
Probability: 1/495
Final answer:

The probability of selecting two defective bulbs is 1/495.

Applied rules:

• Without replacement: Sample space decreases after each selection

• Dependent events: Second probability depends on first outcome

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

8 Multiple Dice Roll
Exercise 8
Two dice are rolled. What is the probability of getting a sum of 7?
Definition:

Sum of dice: When rolling two dice, there are multiple ways to achieve the same sum.

Step 1: List all possible outcomes for sum of 7

(1,6), (2,5), (3,4), (4,3), (5,2), (6,1) = 6 outcomes

Step 2: Count total possible outcomes

6 × 6 = 36 total outcomes

Step 3: Calculate probability

P(sum of 7) = 6/36 = 1/6

Favorable outcomes: 6
÷
Total outcomes: 36
= 1/6
Probability: 1/6
Final answer:

The probability of rolling a sum of 7 is 1/6.

Applied rules:

• Sample space: All possible outcomes of an experiment

• Favorable outcomes: Outcomes that satisfy the condition

• Basic probability: P = Favorable outcomes / Total outcomes

9 Birthday Problem
Exercise 9
What is the probability that two randomly selected people have the same birthday (ignoring leap years)?
Definition:

Birthday problem: A classic probability problem examining the likelihood of shared birthdays.

Step 1: Find probability that second person has same birthday as first

There is 1 day out of 365 that matches the first person's birthday

Step 2: Calculate the probability

P(same birthday) = 1/365

Step 3: State the answer

The probability is 1/365

Matching day: 1
÷
Total days: 365
= 1/365
Probability: 1/365
Final answer:

The probability that two randomly selected people share the same birthday is 1/365.

Applied rules:

• Basic probability: P = Favorable outcomes / Total outcomes

• Fixed event: First person's birthday is fixed

• Simple calculation: Only one favorable outcome out of 365 possibilities

10 Gender Probability
Exercise 10
A family has three children. What is the probability that all three are girls?
Definition:

Multiple births: Each birth is an independent event with equal probability of boy or girl.

Step 1: Find probability of one child being a girl

P(girl) = 1/2

Step 2: Since each birth is independent, multiply probabilities

P(three girls) = (1/2) × (1/2) × (1/2) = 1/8

Step 3: State the answer

The probability is 1/8

First child: 1/2
×
Second child: 1/2
×
Third child: 1/2
= 1/8
Probability: 1/8
Final answer:

The probability that all three children are girls is 1/8.

Applied rules:

• Independent events: Each birth is independent of others

• Equal probability: P(girl) = P(boy) = 1/2

• Multiplication rule: For independent events, multiply individual probabilities

Comprehensive Summary: Compound Probability
Core Concepts & Definitions:

Probability: A measure of the likelihood of an event occurring, expressed as a number between 0 and 1.

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 outcome of another event.

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

With Replacement: The item is returned to the sample space before the next event occurs.

Without Replacement: The item is not returned to the sample space before the next event occurs.

Tree Diagram: A visual representation showing all possible outcomes of sequential events.

Sample Space: The set of all possible outcomes of an experiment.

Core Rules & Formulas:

Essential Formulas:

  • Independent events: P(A and B) = P(A) × P(B)
  • Dependent events: P(A and B) = P(A) × P(B|A)
  • Addition rule: P(A or B) = P(A) + P(B) - P(A and B)
  • Complement rule: P(not A) = 1 - P(A)

Key Rules:

  • Probability is always between 0 and 1 (inclusive)
  • Sum of all probabilities in a sample space equals 1
  • For independent events, multiply probabilities
  • For dependent events, adjust the second probability based on the first outcome
Step-by-Step Process:
  1. Identify events: Determine what events are occurring
  2. Determine independence: Check if events are independent or dependent
  3. Choose formula: Select appropriate probability formula
  4. Calculate individual probabilities: Find probability of each event
  5. Apply formula: Perform the required mathematical operation
  6. Simplify: Reduce the fraction if possible
  7. Verify: Check that answer is between 0 and 1
Examples & Applications:

Simple Independent Example:

  • Rolling a die and flipping a coin: P(3 and heads) = (1/6) × (1/2) = 1/12

Dependent Example:

  • Drawing cards without replacement: P(heart then spade) = (13/52) × (13/51)

Addition Rule Example:

  • P(rolling even or prime) = P(even) + P(prime) - P(even and prime)

Tree Diagram Example:

  • Flipping coin twice: HH, HT, TH, TT each with probability 1/4
Tips, Tricks & Common Pitfalls:

Tips & Tricks:

  • Always identify if events are independent or dependent first
  • Use tree diagrams for complex sequential events
  • Check that your answer is between 0 and 1
  • Remember: with replacement = independent, without replacement = dependent
  • For "at least one" problems, use complement rule: P(at least one) = 1 - P(none)

Common Pitfalls:

  • Confusing independent with dependent events
  • Using multiplication rule for dependent events without adjustment
  • Forgetting to adjust probabilities when sampling without replacement
  • Adding probabilities instead of multiplying for "and" events
Key Notes for Memorization:
  • Independent: P(A and B) = P(A) × P(B)
  • Dependent: P(A and B) = P(A) × P(B|A)
  • Range: 0 ≤ P(event) ≤ 1
  • Sum: P(all outcomes) = 1
  • With replacement: Independent events
  • Without replacement: Dependent events
  • Tree diagrams: Visualize all possible outcomes
Additional Compound Probability Practice
P(A and B) = P(A) × P(B|A)
Conditional Probability Formula
Key definitions:

Compound probability: Probability of multiple events occurring together

Independence: Events that don't affect each other's outcomes

Dependence: Events where one affects the other's probability

Probability methodology:
  1. Identify: Determine the events involved
  2. Classify: Determine if events are independent or dependent
  3. Select: Choose the appropriate probability formula
  4. Calculate: Compute individual probabilities
  5. Combine: Apply the probability formula
  6. Verify: Ensure answer is reasonable (0 ≤ P ≤ 1)
Tip 1: Always include units and simplify fractions when possible.
Tip 2: Draw tree diagrams to visualize complex scenarios.
Tip 3: Check that probabilities are between 0 and 1.
Tip 4: Remember to adjust probabilities for dependent events.
Common errors: Wrong formula application, ignoring dependence, calculation mistakes.
Success strategies: Systematic approach, verification, clear notation.
Essential concepts:

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

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

• Range: 0 ≤ P ≤ 1

• Total: P(Sample Space) = 1

Questions & Answers

Question: How do I know if events are independent or dependent?

Answer: Look for these clues:

  • Independent: The first event does not change the sample space for the second event (e.g., flipping coins, rolling dice, drawing with replacement)
  • Dependent: The first event changes the sample space for the second event (e.g., drawing without replacement, selecting people from a group)

Ask yourself: "Does the first event affect the second event's probability?" If yes, they're dependent.

Question: When do I add probabilities versus multiply them?

Answer: Use this guide:

  • Multiply: For "AND" events (both A and B happen)
  • Add: For "OR" events (either A or B happens)

For "AND" events: P(A and B) = P(A) × P(B) for independent events. For "OR" events: P(A or B) = P(A) + P(B) - P(A and B).

Question: What's the difference between theoretical and experimental probability?

Answer: The differences are:

  • Theoretical probability: Calculated based on possible outcomes (what should happen)
  • Experimental probability: Based on actual trials or observations (what did happen)

Example: Theoretical probability of heads = 1/2, but experimental probability might be 48/100 after 100 coin flips.