Solved Exercises on Theoretical Probability in Grade 7

Master theoretical probability: calculating probabilities, sample spaces, favorable outcomes, and probability rules through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Rolling a Die
Exercise 1
A fair six-sided die is rolled. What is the probability of rolling an even number?
Definition:

Theoretical Probability: P(A) = Number of favorable outcomes / Total number of possible outcomes

Probability calculation method:
  1. Identify the sample space (all possible outcomes)
  2. Count the favorable outcomes
  3. Apply the probability formula
  4. Simplify the fraction if possible
Sample Space
{1, 2, 3, 4, 5, 6}
Favorable Outcomes
{2, 4, 6}
Probability
3/6 = 1/2
Step 1: Identify the sample space

When rolling a standard die, possible outcomes are: 1, 2, 3, 4, 5, 6

Total number of possible outcomes = 6

Step 2: Count favorable outcomes

Even numbers on a die: 2, 4, 6

Number of favorable outcomes = 3

Step 3: Apply probability formula

P(even number) = Number of favorable outcomes / Total number of outcomes

P(even number) = 3/6 = 1/2

Step 4: Express in different forms

1/2 = 0.5 = 50%

P(even number) = 1/2 or 50%
Final answer:

The probability of rolling an even number is 1/2 or 50%

Applied rules:

• Probability Formula: P(A) = Favorable outcomes / Total outcomes

• Sample Space: Complete set of all possible outcomes

• Fraction Simplification: Reduce to lowest terms when possible

2 Flipping a Coin
Exercise 2
A fair coin is flipped twice. What is the probability of getting exactly one head?
Definition:

Compound Events: Multiple events combined using AND/OR operations

Sample Space
{HH, HT, TH, TT}
Favorable Outcomes
{HT, TH}
Probability
2/4 = 1/2
Step 1: List the sample space

When flipping a coin twice, possible outcomes are:

HH (Head, Head), HT (Head, Tail), TH (Tail, Head), TT (Tail, Tail)

Total number of outcomes = 4

Step 2: Identify favorable outcomes

Outcomes with exactly one head: HT, TH

Number of favorable outcomes = 2

Step 3: Calculate probability

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

Step 4: Alternative method

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

P(exactly one head) = 1/2 or 50%
Final answer:

The probability of getting exactly one head is 1/2 or 50%

Applied rules:

• Sample Space for Compound Events: List all possible combinations

• Addition Rule: P(A or B) = P(A) + P(B) for mutually exclusive events

• Independence: Each flip is independent of previous flips

3 Drawing Marbles
Exercise 3
A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. What is the probability of drawing a blue marble?
Definition:

Probability Distribution: Sum of all probabilities equals 1

Total Marbles
5 + 3 + 2 = 10
Blue Marbles
3
Probability
3/10
Step 1: Count total marbles

Red marbles: 5, Blue marbles: 3, Green marbles: 2

Total marbles = 5 + 3 + 2 = 10

Step 2: Identify favorable outcomes

Number of blue marbles = 3

Step 3: Apply probability formula

P(blue marble) = Number of blue marbles / Total number of marbles

P(blue marble) = 3/10

Step 4: Express in different forms

3/10 = 0.3 = 30%

P(blue marble) = 3/10 or 30%
Final answer:

The probability of drawing a blue marble is 3/10 or 30%

Applied rules:

• Probability Formula: P(A) = Favorable outcomes / Total outcomes

• Counting Principle: Add up all possible items to get total

• Probability Range: Probabilities are between 0 and 1 (or 0% and 100%)

Rules and methods, laws,...
\(P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\)
Theoretical Probability Formula
Range
\(0 \leq P(A) \leq 1\)
Probability is between 0 and 1
Complement
\(P(A') = 1 - P(A)\)
Probability of complement event
Sample Space
\(P(S) = 1\)
Probability of sample space is 1
Key definitions:

Sample Space: Set of all possible outcomes of an experiment

Event: Subset of the sample space

Favorable Outcome: Outcome that satisfies the condition

Theoretical Probability: Calculated probability based on theory

Complete methodology:
  1. Define the experiment: Understand what is being tested
  2. List the sample space: Identify all possible outcomes
  3. Identify the event: Determine what outcome(s) are favorable
  4. Count outcomes: Count favorable and total outcomes
  5. Calculate probability: Apply the formula
  6. Simplify: Reduce fraction if possible
Tip 1: Always check that your probability is between 0 and 1.
Tip 2: The sum of all individual probabilities in a sample space equals 1.
Tip 3: For "at least" problems, often easier to use complement rule.
Tip 4: Draw tree diagrams for compound events to visualize outcomes.
Common errors: Forgetting to count all possible outcomes, miscounting favorable outcomes, not simplifying fractions.
Exam preparation: Practice listing sample spaces, memorize probability formulas, work on word problems.
Formulas to know by heart:

• Basic probability: P(A) = favorable outcomes / total outcomes

• Complement rule: P(not A) = 1 - P(A)

• Addition rule: P(A or B) = P(A) + P(B) - P(A and B)

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

• Sample space: P(S) = 1

Solution: Exercises 4 to 5
4 Spinner Problem
Exercise 4
A spinner has 8 equal sections numbered 1 through 8. What is the probability of spinning a number greater than 5?
Definition:

Uniform Probability: All outcomes equally likely in fair experiments

Sample Space
{1, 2, 3, 4, 5, 6, 7, 8}
Favorable Outcomes
{6, 7, 8}
Probability
3/8
Step 1: Identify the sample space

Spinner has 8 equal sections: {1, 2, 3, 4, 5, 6, 7, 8}

Total number of outcomes = 8

Step 2: Find numbers greater than 5

Numbers greater than 5: 6, 7, 8

Number of favorable outcomes = 3

Step 3: Calculate probability

P(number > 5) = 3/8

Step 4: Express in decimal and percentage

3/8 = 0.375 = 37.5%

P(number > 5) = 3/8 or 37.5%
Final answer:

The probability of spinning a number greater than 5 is 3/8 or 37.5%

Applied rules:

• Set Theory: Identify subset of favorable outcomes

• Comparison: Carefully compare values to determine condition

• Equality: Equal sections mean equal probability for each outcome

5 Card Selection
Exercise 5
A standard deck of 52 cards is shuffled. What is the probability of drawing a face card (Jack, Queen, or King)?
Definition:

Face Cards: Jack, Queen, and King of each suit (12 total in a deck)

Total Cards
52
Face Cards
12
Probability
12/52 = 3/13
Step 1: Understand the deck composition

A standard deck has 52 cards: 4 suits (hearts, diamonds, clubs, spades) × 13 ranks each

Step 2: Count face cards

Each suit has 3 face cards: Jack, Queen, King

Total face cards = 4 suits × 3 face cards per suit = 12

Step 3: Apply probability formula

P(face card) = Number of face cards / Total number of cards

P(face card) = 12/52

Step 4: Simplify the fraction

12/52 = (12÷4)/(52÷4) = 3/13

Step 5: Convert to decimal and percentage

3/13 ≈ 0.231 = 23.1%

P(face card) = 3/13 ≈ 23.1%
Final answer:

The probability of drawing a face card is 3/13 or approximately 23.1%

Applied rules:

• Multiplication: Use counting principle for groups within groups

• Fraction Reduction: Always simplify fractions to lowest terms

• Card Knowledge: Know standard deck composition (52 cards, 4 suits, 13 ranks)

Key Concepts: Probability Fundamentals, Rules, and Applications
\(P(A) = \frac{n(A)}{n(S)}\)
Basic Probability Formula
Key definitions:

Experiment: Process that leads to observable outcomes

Outcome: Result of a single trial of an experiment

Event: Collection of one or more outcomes

Sample Space: Set of all possible outcomes of an experiment

Theoretical Probability: Probability calculated using mathematical reasoning

Complete methodology:
  1. Identify the experiment: Determine what action is being performed
  2. List all possible outcomes: Create the sample space
  3. Determine favorable outcomes: Identify which outcomes satisfy the condition
  4. Count outcomes: Count both favorable and total outcomes
  5. Apply formula: Use P(A) = favorable outcomes / total outcomes
  6. Simplify and interpret: Reduce fraction and express meaningfully
Tip 1: Always ensure the sample space includes ALL possible outcomes.
Tip 2: For complex events, break them down into simpler components.
Tip 3: Use the complement rule when direct calculation is difficult.
Tip 4: Draw Venn diagrams or tree diagrams to visualize relationships.
Common errors: Not considering all outcomes, miscounting, forgetting to simplify fractions, confusing experimental and theoretical probability.
Real-world applications: Weather forecasting, game design, quality control, medical testing, risk assessment.
Essential formulas and rules:

• Basic probability: P(A) = n(A)/n(S) where n(A) is number of favorable outcomes

• Complement: P(A') = 1 - P(A)

• Addition rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

• Multiplication rule: P(A ∩ B) = P(A) × P(B|A) for dependent events

• For independent events: P(A ∩ B) = P(A) × P(B)

• Sample space probability: P(S) = 1

• Impossible event: P(∅) = 0

Visualizing Probability: Different Scenarios
Exercise 6: Probability Comparisons
Compare the theoretical probabilities of various scenarios:
1. Rolling a 6 on a fair die
2. Drawing an Ace from a deck of cards
3. Getting heads on a coin flip
4. Drawing a red card from a deck

Analysis: The chart compares different probability scenarios showing their relative likelihoods.

  • Rolling a 6: 1/6 ≈ 16.7%
  • Getting heads: 1/2 = 50%
  • Drawing an Ace: 4/52 = 1/13 ≈ 7.7%
  • Drawing a red card: 26/52 = 1/2 = 50%

Questions & Answers

Question: I don't understand the difference between theoretical probability and experimental probability. Can you explain?

Answer: Great question! Here's the key distinction:

  • Theoretical Probability: Calculated using mathematical reasoning based on what SHOULD happen. Example: The probability of rolling a 3 on a fair die is 1/6 because there's 1 favorable outcome out of 6 possible outcomes.
  • Experimental Probability: Based on actual results from performing an experiment multiple times. Example: If you roll a die 60 times and get a 3 ten times, the experimental probability is 10/60 = 1/6.

As the number of trials increases, experimental probability tends to approach theoretical probability. Theoretical probability is predictive, while experimental probability is based on observed data.

In your grade 7 studies, you'll focus mainly on theoretical probability calculations using the formula: P(A) = favorable outcomes / total outcomes.

Question: When calculating probability, how do I know if I've listed all the possible outcomes correctly?

Answer: Here are strategies to ensure you've identified all possible outcomes:

  • Systematic Approach: List outcomes in order (1, 2, 3, 4, 5, 6 for a die) or use a tree diagram
  • Count Verification: For standard objects, know the totals (die: 6 sides, coin: 2 sides, deck: 52 cards)
  • Pattern Recognition: For compound events, multiply individual possibilities (coin flip × die roll = 2 × 6 = 12 outcomes)
  • Double Check: Ask yourself: "Could anything else possibly happen?"

Example: When rolling two dice, systematically list outcomes as ordered pairs: (1,1), (1,2), ..., (6,6) giving 36 total outcomes.

Always verify that the sum of all individual probabilities equals 1.

Question: Why do we simplify fractions when expressing probability? Is it wrong to leave them unsimplified?

Answer: While leaving fractions unsimplified isn't mathematically incorrect, it's considered bad practice for several reasons:

  • Clarity: Simplified fractions make it easier to understand the relationship between favorable and total outcomes
  • Standard Form: Mathematical convention expects fractions in lowest terms
  • Comparison: It's easier to compare 1/2 vs 3/6 when both are simplified
  • Communication: Simplified fractions are more efficient to communicate and understand

Example: 6/12 and 1/2 represent the same probability, but 1/2 clearly shows 1 favorable outcome out of 2 possible categories.

Always reduce fractions to lowest terms unless specifically instructed otherwise!