Solved Exercises on Simple Probability in Grade 7

Master simple probability: calculating likelihood, sample spaces, and theoretical probability through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Rolling a Die
Exercise 1
What is the probability of rolling a 4 on a standard six-sided die?
Definition:

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

Formula: P(Event) = (Number of favorable outcomes) ÷ (Total number of possible outcomes)

Probability Calculation Method:
  1. Identify the event of interest
  2. Determine the number of favorable outcomes
  3. Determine the total number of possible outcomes
  4. Apply the probability formula
  5. Simplify the fraction if possible
Favorable Outcomes
1
Total Outcomes
6
Probability
1/6
Step 1: Identify the event

Event: Rolling a 4

Step 2: Count favorable outcomes

On a standard die, there is only 1 way to roll a 4

Number of favorable outcomes = 1

Step 3: Count total possible outcomes

A standard die has 6 faces: 1, 2, 3, 4, 5, 6

Total possible outcomes = 6

Step 4: Apply the probability formula

P(Rolling a 4) = Favorable outcomes ÷ Total outcomes

P(Rolling a 4) = 1 ÷ 6 = 1/6

P(Rolling a 4) = 1/6
Final Answer:

The probability of rolling a 4 on a standard six-sided die is 1/6.

Applied Rules:

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

• Standard Die: Has 6 faces numbered 1-6

• Range: Probability values are between 0 and 1

2 Spinner Probability
Exercise 2
A spinner has 8 equal sections numbered 1 through 8. What is the probability of landing on an even number?
Definition:

Even Numbers: Integers divisible by 2 (2, 4, 6, 8, 10, ...)

Spinner: A circular device divided into equal sections used for probability experiments.

Even Numbers
2, 4, 6, 8
Favorable Outcomes
4
Total Outcomes
8
Probability
1/2
Step 1: Identify the event

Event: Landing on an even number

Step 2: List all possible outcomes

Spinner sections: 1, 2, 3, 4, 5, 6, 7, 8

Total possible outcomes = 8

Step 3: Identify favorable outcomes

Even numbers: 2, 4, 6, 8

Number of favorable outcomes = 4

Step 4: Apply the probability formula

P(Even number) = Favorable outcomes ÷ Total outcomes

P(Even number) = 4 ÷ 8 = 1/2

P(Even number) = 1/2
Final Answer:

The probability of landing on an even number is 1/2.

Applied Rules:

• Even Numbers: Divisible by 2

• Equal Sections: Each section has equal probability

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

3 Marble Bag
Exercise 3
A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. What is the probability of randomly selecting a blue marble?
Definition:

Marble Bag: A container holding colored marbles used for probability experiments.

Random Selection: Each item has an equal chance of being selected.

Blue Marbles
3
Total Marbles
10
Probability
3/10
Step 1: Identify the event

Event: Selecting a blue marble

Step 2: Count favorable outcomes

Number of blue marbles = 3

Step 3: Count total possible outcomes

Total marbles = 5 red + 3 blue + 2 green = 10 marbles

Step 4: Apply the probability formula

P(Blue marble) = Blue marbles ÷ Total marbles

P(Blue marble) = 3 ÷ 10 = 3/10

Step 5: Express as decimal or percentage

3/10 = 0.3 = 30%

P(Blue marble) = 3/10 = 0.3 = 30%
Final Answer:

The probability of randomly selecting a blue marble is 3/10 or 30%.

Applied Rules:

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

• Total Count: Sum all possible outcomes

• Equivalent Forms: Fraction, decimal, and percentage are equivalent

Rules and methods, laws,...
\(P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\)
Probability Formula
Probability Range
\(0 \leq P(E) \leq 1\)
Between 0 and 1
Certain Event
\(P(E) = 1\)
Will definitely happen
Impossible Event
\(P(E) = 0\)
Will never happen
Key Definitions:

Probability: A measure of the likelihood that an event will occur

Event: A specific outcome or set of outcomes in a probability experiment

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

Favorable Outcomes: The outcomes that satisfy the condition of the event

Impossible Event: An event that has no chance of occurring (probability = 0)

Certain Event: An event that will definitely occur (probability = 1)

Simple Probability: Probability of a single event with equally likely outcomes

Complete Methodology:
  1. Identify the Experiment: Understand what is happening (rolling dice, spinning wheel, drawing cards)
  2. Define the Event: Specify exactly what you want to find the probability of
  3. Count Total Outcomes: Determine the total number of possible outcomes
  4. Count Favorable Outcomes: Determine how many outcomes satisfy the event
  5. Apply Formula: Calculate P(E) = (favorable outcomes) ÷ (total outcomes)
  6. Simplify: Reduce the fraction to lowest terms if possible
Tip 1: Always list all possible outcomes to ensure accuracy.
Tip 2: Probability values are always between 0 and 1 (or 0% and 100%).
Tip 3: For multiple favorable outcomes, add them up.
Tip 4: Convert between fractions, decimals, and percentages as needed.
Tip 5: Verify that your answer makes sense in the context of the problem.
Common Errors: Miscounting outcomes, confusing favorable with total, forgetting to simplify fractions, miscalculating probabilities.
Exam Preparation: Practice with various scenarios, memorize the formula, understand sample spaces, know how to interpret results.
Solution: Exercises 4 to 5
4 Card Deck
Exercise 4
A standard deck of 52 cards is shuffled. What is the probability of drawing a heart? What is the probability of drawing a face card (Jack, Queen, King)?
Definition:

Standard Deck: 52 cards with 4 suits (hearts, diamonds, clubs, spades) of 13 cards each.

Face Cards: Jack, Queen, and King of each suit (3 per suit).

Heart Probability
13/52 = 1/4
Face Card Probability
12/52 = 3/13
Step 1: Identify total outcomes

Standard deck has 52 cards

Total possible outcomes = 52

Step 2: Calculate heart probability

Hearts: 13 cards (Ace through King)

Favorable outcomes = 13

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

Step 3: Calculate face card probability

Face cards per suit: Jack, Queen, King = 3

Number of suits: 4

Total face cards = 3 × 4 = 12

P(Face card) = 12/52 = 3/13

Step 4: Verify calculations

Hearts: 13/52 = 1/4 = 0.25 = 25%

Face cards: 12/52 = 3/13 ≈ 0.231 = 23.1%

P(Heart) = 1/4, P(Face card) = 3/13
Final Answer:

The probability of drawing a heart is 1/4 or 25%. The probability of drawing a face card is 3/13 or approximately 23.1%.

Applied Rules:

• Standard Deck: 52 cards, 4 suits of 13 each

• Face Cards: 3 per suit, 12 total

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

5 Real-World Application
Exercise 5
In a class of 30 students, 18 are girls and 12 are boys. If a student is selected at random, what is the probability that the student is a girl? What is the probability that the student is a boy?
Definition:

Real-World Probability: Applying probability concepts to practical situations.

Random Selection: Each individual has an equal chance of being selected.

Girl Probability
18/30 = 3/5
Boy Probability
12/30 = 2/5
Verification
3/5 + 2/5 = 1
Step 1: Identify total outcomes

Total students = 30

Total possible outcomes = 30

Step 2: Calculate girl probability

Number of girls = 18

Favorable outcomes for girls = 18

P(Girl) = 18/30 = 3/5

Step 3: Calculate boy probability

Number of boys = 12

Favorable outcomes for boys = 12

P(Boy) = 12/30 = 2/5

Step 4: Verify the probabilities

P(Girl) + P(Boy) = 3/5 + 2/5 = 5/5 = 1

This confirms our calculations are correct

Step 5: Express as percentages

P(Girl) = 3/5 = 0.6 = 60%

P(Boy) = 2/5 = 0.4 = 40%

P(Girl) = 3/5 = 60%, P(Boy) = 2/5 = 40%
Final Answer:

The probability of selecting a girl is 3/5 or 60%. The probability of selecting a boy is 2/5 or 40%.

Applied Rules:

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

• Complementary Events: P(Girl) + P(Boy) = 1

• Real-World Context: Apply probability to practical situations

Detailed Summary: Simple Probability Fundamentals
\(P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}\)
Probability Formula
Key definitions:

Probability: A measure of the likelihood that an event will occur, expressed as a number between 0 and 1

Event: A specific outcome or set of outcomes in a probability experiment

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

Favorable Outcomes: The outcomes that satisfy the condition of the event

Simple Probability: The probability of a single event with equally likely outcomes

Impossible Event: An event that has no chance of occurring (probability = 0)

Certain Event: An event that will definitely occur (probability = 1)

Complete methodology:
  1. Experiment Identification: Clearly define what is happening in the probability scenario
  2. Event Definition: Specify exactly what outcome you're looking for
  3. Outcome Counting: Count the total number of possible outcomes
  4. Favorable Outcome Counting: Count how many outcomes satisfy the event condition
  5. Formula Application: Apply P(E) = (favorable outcomes) ÷ (total outcomes)
  6. Result Expression: Express as fraction, decimal, or percentage as appropriate
Tip 1: Always list all possible outcomes to ensure you don't miss any.
Tip 2: Remember that probability values are always between 0 and 1 inclusive.
Tip 3: For multiple favorable outcomes, count them all separately.
Tip 4: Check that your answer makes sense by considering the context.
Tip 5: Remember that all probabilities in a sample space sum to 1.
Common errors: Miscounting outcomes, confusing favorable with total, forgetting to simplify fractions, miscalculating, not considering all possible outcomes.
Exam preparation: Practice with various scenarios, memorize the formula, understand sample spaces, know how to convert between forms, verify your answers.
Formulas to know by heart:

• Probability Formula: P(E) = (Number of favorable outcomes) ÷ (Total number of possible outcomes)

• Probability Range: 0 ≤ P(E) ≤ 1

• Complementary Probability: P(not E) = 1 - P(E)

• Certain Event: P(E) = 1

• Impossible Event: P(E) = 0

Exercise with Visualization: Probability Distributions
Exercise 6: Comparing Probabilities
Compare the probabilities of different events:
Event A: Rolling an even number on a die
Event B: Drawing a heart from a deck
Event C: Flipping heads on a coin

Analysis: The visualization shows different probability values for various events.

  • Event A: P(Even) = 3/6 = 1/2 = 50%
  • Event B: P(Heart) = 13/52 = 1/4 = 25%
  • Event C: P(Head) = 1/2 = 50%

Questions & Answers

Question: How do I know if I've counted all the possible outcomes correctly?

Answer: Here are effective strategies to ensure you count all outcomes correctly:

  • Systematic Listing: List outcomes in a logical order (1, 2, 3, 4, 5, 6 for a die)
  • Visual Representation: Draw pictures or use diagrams when possible
  • Double-Check Counting: Count forwards and backwards to verify
  • Grouping Strategy: Group similar outcomes together
  • Context Understanding: Make sure your count matches the problem description

For example, with a die: 1, 2, 3, 4, 5, 6 - count them: 1, 2, 3, 4, 5, 6 = 6 outcomes.

For a deck of cards: 4 suits × 13 cards each = 52 total cards.

Always verify that your count makes sense in the context of the problem!

Question: Can probability ever be greater than 1 or less than 0? Why or why not?

Answer: No, probability can never be greater than 1 or less than 0. Here's why:

  • Lower Bound (0): Impossible events have 0 favorable outcomes, so P = 0÷total = 0
  • Upper Bound (1): Certain events have all outcomes as favorable, so P = total÷total = 1
  • Logic: You can't have more favorable outcomes than total possible outcomes
  • Definition: Probability is defined as a ratio where favorable ≤ total

If you calculate a probability greater than 1 or less than 0, you made an error:

  • You may have miscounted favorable outcomes
  • You may have miscounted total outcomes
  • You may have divided incorrectly

Always check that your answer is between 0 and 1 (or 0% and 100%)!

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

Answer: Here are several verification methods:

  1. Range Check: Ensure the probability is between 0 and 1
  2. Reasonableness Check: Does the probability make sense given the situation?
  3. Count Verification: Double-check your counts of favorable and total outcomes
  4. Alternative Calculation: Try calculating using a different approach
  5. Complement Check: P(E) + P(not E) should equal 1

Example: If P(rolling a 4) = 1/6, then P(not rolling a 4) = 5/6.

Check: 1/6 + 5/6 = 6/6 = 1 ✓

These checks help catch calculation errors and ensure accuracy!

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

Answer: The key differences are:

  • Theoretical Probability: Calculated using mathematical principles and counting
  • Experimental Probability: Determined by actually performing trials and recording results

Theoretical Example: P(heads) = 1/2 (mathematical calculation)

Experimental Example: Flip a coin 100 times, get 52 heads → P(heads) = 52/100 = 0.52

With more trials, experimental probability approaches theoretical probability.

In this chapter, we focus on theoretical probability where we calculate based on the structure of the experiment.

Question: When would I use probability in real life?

Answer: Probability is used in many real-life situations:

  • Weather Forecasting: Probability of rain, snow, or sunshine
  • Games: Odds in card games, board games, or sports
  • Insurance: Calculating risk and premiums
  • Medicine: Success rates of treatments or surgeries
  • Business: Predicting sales, market trends, or success of products
  • Quality Control: Testing products for defects

Understanding probability helps you make informed decisions and assess risks in daily life.

For example, knowing that there's a 20% chance of rain helps you decide whether to bring an umbrella!