Solved Exercises on Probability Models in Grade 7

Master probability models: creating sample spaces, calculating compound probabilities, and modeling real-world scenarios through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Sample Space Creation
Exercise 1
Create a sample space for flipping a coin and rolling a six-sided die. How many possible outcomes are there?
Definition:

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

Compound Event: An event that combines two or more simple events.

Tree Diagram: A visual representation of all possible outcomes.

Sample Space Creation Method:
  1. Identify all possible outcomes for each simple event
  2. Combine outcomes systematically
  3. List all possible combinations
  4. Count the total number of outcomes
  5. Verify using the multiplication principle
Coin Outcomes
H, T
Die Outcomes
1, 2, 3, 4, 5, 6
Total Outcomes
2 × 6 = 12
Step 1: Identify outcomes for each event

Coin flip: Heads (H) or Tails (T) = 2 outcomes

Die roll: 1, 2, 3, 4, 5, 6 = 6 outcomes

Step 2: Create systematic combinations

For each coin outcome, pair with each die outcome:

H1, H2, H3, H4, H5, H6

T1, T2, T3, T4, T5, T6

Step 3: List complete sample space

S = {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}

Step 4: Count total outcomes

Using multiplication principle: 2 × 6 = 12 outcomes

Sample Space: {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}
Total Outcomes: 12
Final Answer:

The sample space consists of 12 outcomes: {H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6}. There are 12 possible outcomes.

Applied Rules:

• Multiplication Principle: If event A has m outcomes and event B has n outcomes, then A and B combined have m × n outcomes

• Systematic Listing: Organize outcomes to avoid missing any

• Sample Space: Must include all possible outcomes

2 Tree Diagram
Exercise 2
Draw a tree diagram for flipping two coins. What is the probability of getting exactly one head?
Definition:

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

Sequential Events: Events that occur in a specific order.

First Coin
H or T
Second Coin
H or T
One Head
HT or TH
Probability
2/4 = 1/2
Step 1: Draw first level of tree

Start with the first coin flip

Branch 1: Heads (H)

Branch 2: Tails (T)

Step 2: Draw second level of tree

From each first coin outcome, branch to second coin outcomes:

From H: H→H (HH), H→T (HT)

From T: T→H (TH), T→T (TT)

Step 3: List all possible outcomes

Sample space: {HH, HT, TH, TT}

Step 4: Identify favorable outcomes

Exactly one head: HT and TH

Number of favorable outcomes = 2

Step 5: Calculate probability

Total outcomes = 4

Favorable outcomes = 2

P(Exactly one head) = 2/4 = 1/2

P(Exactly one head) = 1/2
Final Answer:

The probability of getting exactly one head when flipping two coins is 1/2 or 50%.

Applied Rules:

• Tree Diagram: Visualize all possible outcomes systematically

• Probability Formula: P(E) = (favorable outcomes) ÷ (total outcomes)

• Sequential Events: Each stage branches from previous outcomes

3 Compound Probability
Exercise 3
A bag contains 3 red marbles and 2 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 before drawing the second.

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

First Draw
P(Red₁) = 3/5
Second Draw
P(Red₂|Red₁) = 2/4
Both Red
P(Red₁ and Red₂) = 6/20 = 3/10
Step 1: Calculate probability of first red marble

Initial state: 3 red, 2 blue, total = 5 marbles

P(First marble is red) = 3/5

Step 2: Calculate probability of second red marble given first was red

After drawing first red marble: 2 red, 2 blue, total = 4 marbles

P(Second marble is red | First was red) = 2/4 = 1/2

Step 3: Apply multiplication rule for dependent events

P(Both red) = P(First red) × P(Second red | First red)

P(Both red) = (3/5) × (1/2) = 3/10

Step 4: Verify using sample space

Sample space for drawing 2 marbles from 5: C(5,2) = 10 pairs

Pairs of red marbles: C(3,2) = 3 pairs

P(Both red) = 3/10 ✓

P(Both marbles are red) = 3/10
Final Answer:

The probability that both marbles drawn are red is 3/10 or 30%.

Applied Rules:

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

• Without Replacement: Total outcomes decrease after each draw

• Conditional Probability: Probability of B given A has occurred

Rules and methods, laws,...
\(P(A \text{ and } B) = P(A) \times P(B|A)\)
Multiplication Rule
Sample Space
Set of all possible outcomes
Complete list of outcomes
Multiplication Principle
m × n outcomes
For independent events
Tree Diagram
Visual representation
Shows all outcomes
Key Definitions:

Probability Model: A mathematical representation of a random phenomenon

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

Event: A subset of the sample space

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

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

Tree Diagram: A visual tool to represent all possible outcomes of sequential events

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

Complete Methodology:
  1. Identify Events: Determine all possible outcomes for each component
  2. Create Sample Space: List all possible combinations systematically
  3. Determine Dependencies: Identify if events are independent or dependent
  4. Apply Probability Rules: Use appropriate formulas based on dependencies
  5. Calculate Probabilities: Compute the desired probability
  6. Verify Results: Check if the answer makes sense in context
Tip 1: Always create a systematic list or diagram to avoid missing outcomes.
Tip 2: For dependent events, adjust the sample space after each draw.
Tip 3: Use tree diagrams for complex sequential events.
Tip 4: Check that all probabilities sum to 1 when appropriate.
Tip 5: Remember that "without replacement" means dependent events.
Common Errors: Forgetting to adjust for dependent events, miscounting outcomes, not using systematic approaches, confusing independent and dependent events.
Exam Preparation: Practice creating sample spaces, master tree diagrams, understand independence vs dependence, memorize probability rules.
Solution: Exercises 4 to 5
4 Spinner Probability Model
Exercise 4
A spinner has 4 equal sections colored red, blue, green, and yellow. The spinner is spun twice. Create a probability model for getting the same color both times. What is this probability?
Definition:

Probability Model: A mathematical description of a random phenomenon.

Same Color: Both spins land on identical colors.

Same Colors
RR, BB, GG, YY
Total Outcomes
4 × 4 = 16
Probability
4/16 = 1/4
Step 1: Create sample space for two spins

First spin: R, B, G, Y

Second spin: R, B, G, Y

Sample space: {RR, RB, RG, RY, BR, BB, BG, BY, GR, GB, GG, GY, YR, YB, YG, YY}

Step 2: Identify favorable outcomes

Same color both times: RR, BB, GG, YY

Number of favorable outcomes = 4

Step 3: Count total outcomes

Total possible outcomes = 4 × 4 = 16

Step 4: Calculate probability

P(Same color) = 4/16 = 1/4

Step 5: Alternative calculation

P(RR) = (1/4) × (1/4) = 1/16

P(BB) = (1/4) × (1/4) = 1/16

P(GG) = (1/4) × (1/4) = 1/16

P(YY) = (1/4) × (1/4) = 1/16

P(Same color) = 1/16 + 1/16 + 1/16 + 1/16 = 4/16 = 1/4

P(Same color) = 1/4
Final Answer:

The probability of getting the same color on both spins is 1/4 or 25%.

Applied Rules:

• Independent Events: Each spin is independent

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

• Addition Rule: For mutually exclusive events, add probabilities

5 Real-World Probability Model
Exercise 5
In a class of 20 students, 12 are girls and 8 are boys. Two students are randomly selected without replacement to be class representatives. What is the probability that both are girls? Create a probability model for this scenario.
Definition:

Real-World Scenario: Applying probability models to practical situations.

Without Replacement: The first selection affects the probability of the second.

First Selection
P(Girl₁) = 12/20
Second Selection
P(Girl₂|Girl₁) = 11/19
Both Girls
P(Girl₁ and Girl₂) = 132/380 = 33/95
Step 1: Identify the scenario

Class: 12 girls, 8 boys, total = 20 students

Selection: 2 students without replacement

Step 2: Calculate probability of first girl

P(First student is a girl) = 12/20 = 3/5

Step 3: Calculate probability of second girl given first was a girl

After selecting first girl: 11 girls, 8 boys, total = 19 students

P(Second student is a girl | First was a girl) = 11/19

Step 4: Apply multiplication rule for dependent events

P(Both girls) = P(First girl) × P(Second girl | First girl)

P(Both girls) = (12/20) × (11/19) = 132/380 = 33/95

Step 5: Express as decimal or percentage

33/95 ≈ 0.347 = 34.7%

Step 6: Verify using combination formula

Ways to select 2 girls from 12: C(12,2) = 66

Ways to select 2 students from 20: C(20,2) = 190

P(Both girls) = 66/190 = 33/95 ✓

P(Both students are girls) = 33/95 ≈ 34.7%
Final Answer:

The probability that both selected students are girls is 33/95 or approximately 34.7%.

Applied Rules:

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

• Without Replacement: Total population decreases after each selection

• Real-World Application: Apply probability models to practical scenarios

Detailed Summary: Probability Models Fundamentals
\(P(A \text{ and } B) = P(A) \times P(B|A)\)
Multiplication Rule for Dependent Events
Key definitions:

Probability Model: A mathematical representation of a random phenomenon that specifies all possible outcomes and their associated probabilities

Sample Space: The set of all possible outcomes of an experiment, denoted as S

Event: A subset of the sample space representing a particular outcome or set of outcomes

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

Dependent Events: Events where the occurrence of one affects the probability of the other

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

Conditional Probability: The probability of an event given that another event has occurred, denoted as P(B|A)

Complete methodology:
  1. Problem Analysis: Identify the random phenomenon and what needs to be calculated
  2. Sample Space Creation: Systematically list all possible outcomes
  3. Event Identification: Determine which outcomes satisfy the condition
  4. Dependency Check: Determine if events are independent or dependent
  5. Rule Application: Apply appropriate probability rules
  6. Calculation: Perform the mathematical computations
  7. Verification: Check that the answer is reasonable and follows probability rules
Tip 1: Always identify if events are independent or dependent before applying formulas.
Tip 2: Use tree diagrams for complex sequential events to visualize all outcomes.
Tip 3: For "without replacement" scenarios, the sample space changes after each draw.
Tip 4: Systematic listing prevents missing outcomes in the sample space.
Tip 5: Always verify that probabilities are between 0 and 1 inclusive.
Common errors: Confusing independent and dependent events, not adjusting for "without replacement", miscounting outcomes, forgetting to use conditional probability for dependent events.
Exam preparation: Practice creating sample spaces, master tree diagrams, understand independence vs dependence, memorize probability rules, practice word problems.
Formulas to know by heart:

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

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

• Sample Space: S = {all possible outcomes}

• Multiplication Principle: m × n for two independent events

• Conditional Probability: P(B|A) = P(A and B) ÷ P(A)

Exercise with Visualization: Probability Model Comparison
Exercise 6: Independent vs Dependent Events
Compare probability models for these scenarios:
Scenario A: Drawing two cards with replacement
Scenario B: Drawing two cards without replacement
Scenario C: Flipping two coins

Analysis: The visualization shows how replacement affects probability calculations.

  • Scenario A: Independent events (probabilities remain constant)
  • Scenario B: Dependent events (probabilities change after first draw)
  • Scenario C: Independent events (coin flips don't affect each other)

Questions & Answers

Question: How do I know if events are independent or dependent? This seems confusing.

Answer: Here's how to identify independence:

  • Independent Events: The outcome of the first event does NOT affect the probability of the second event
  • Dependent Events: The outcome of the first event DOES affect the probability of the second event

Examples of Independent Events:

  • Flipping two coins (first flip doesn't affect second)
  • Rolling two dice (first roll doesn't affect second)
  • Drawing cards WITH replacement

Examples of Dependent Events:

  • Drawing marbles WITHOUT replacement
  • Selecting students from a class without replacement
  • Dealing cards from a deck

Ask yourself: "Does the first event change the situation for the second event?" If yes, they're dependent!

Question: When should I use a tree diagram versus just listing the sample space?

Answer: Use tree diagrams when:

  • Events happen in sequence (step by step)
  • You need to track how one event affects another
  • There are multiple stages to the experiment
  • The sample space is complex or large
  • You're dealing with conditional probabilities

List the sample space when:

  • Events happen simultaneously
  • The sample space is small and manageable
  • You just need to count total outcomes
  • Events are independent

Example: Use a tree diagram for "drawing two marbles without replacement" because the first draw affects the second. Use a simple list for "flipping two coins" since outcomes don't affect each other.

Question: What's the difference between "with replacement" and "without replacement" and how does it affect probability?

Answer: The key differences are:

With Replacement:

  • The item is put back after being selected
  • Probabilities remain the same for each draw
  • Events are independent
  • Example: Drawing a card, noting it, putting it back, then drawing again

Without Replacement:

  • The item is NOT put back after being selected
  • Probabilities change after each draw
  • Events are dependent
  • Example: Drawing two cards without putting the first back

For example, drawing two red marbles from 3 red and 2 blue:

  • With replacement: P(both red) = (3/5) × (3/5) = 9/25
  • Without replacement: P(both red) = (3/5) × (2/4) = 6/20 = 3/10

The "without replacement" probability is lower because after drawing one red marble, there are fewer red marbles left!

Question: How can I check if my probability model is correct?

Answer: Here are verification methods:

  1. Range Check: Ensure all probabilities are between 0 and 1
  2. Sum Check: For a complete sample space, all probabilities should sum to 1
  3. Logic Check: Does the probability make sense in context?
  4. Alternative Method: Try calculating the same probability using a different approach
  5. Count Verification: Double-check your counting of outcomes

Example: For flipping two coins, sample space = {HH, HT, TH, TT}

  • Each outcome has probability 1/4
  • Sum: 1/4 + 1/4 + 1/4 + 1/4 = 1 ✓
  • All probabilities between 0 and 1 ✓

Always verify that your model makes logical sense!

Question: When would I encounter probability models in real life?

Answer: Probability models are used everywhere:

  • Games: Calculating odds in card games, board games, or lotteries
  • Business: Risk assessment, quality control, market predictions
  • Medicine: Success rates of treatments, disease transmission
  • Weather: Probability of rain, storms, or other conditions
  • Insurance: Calculating premiums based on risk models
  • Sports: Player performance predictions, game outcomes

For example, a store manager might use a probability model to predict how many customers will buy a product based on past sales data. A doctor might use probability models to determine the effectiveness of a treatment.

Understanding probability models helps you make better decisions in uncertain situations!