Solved Exercises on Simple Probability in Grade 8

Master simple probability: theoretical probability, sample spaces, and event outcomes 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:

Simple Probability: The likelihood of a single event occurring, calculated as favorable outcomes divided by total possible outcomes

Probability calculation method:
  1. Identify the total number of possible outcomes
  2. Identify the number of favorable outcomes
  3. Apply the probability formula
  4. Simplify the fraction if possible
Possible Outcomes
1,2,3,4,5,6
Favorable Outcomes
Rolling a 4
Probability
1/6
Step 1: Identify total possible outcomes

A standard die has 6 faces numbered 1 through 6

Total possible outcomes = 6

Step 2: Identify favorable outcomes

We want to roll a 4, so there is only 1 favorable outcome

Favorable outcomes = 1

Step 3: Apply the probability formula

Probability = Favorable outcomes ÷ Total outcomes

Probability = 1 ÷ 6 = 1/6

P(rolling a 4) = 1/6
Final answer:

The probability of rolling a 4 is 1/6

Applied rules:

• Formula: P(event) = (favorable outcomes)/(total outcomes)

• Range: Probability is always between 0 and 1

• Equally likely: Each outcome has equal chance

2 Flipping a Coin
Exercise 2
What is the probability of getting heads when flipping a fair coin?
Definition:

Fair Coin: A coin with equal probability of landing on heads or tails

Possible Outcomes
Heads, Tails
Favorable Outcomes
Heads
Probability
1/2
Step 1: Identify total possible outcomes

A coin has 2 sides: heads and tails

Total possible outcomes = 2

Step 2: Identify favorable outcomes

We want heads, so there is 1 favorable outcome

Favorable outcomes = 1

Step 3: Apply the probability formula

Probability = 1 ÷ 2 = 1/2

P(heads) = 1/2
Final answer:

The probability of getting heads is 1/2

Applied rules:

• Fair coin: P(heads) = P(tails) = 1/2

• Complement: P(heads) + P(tails) = 1

• Equally likely: Both outcomes have equal probability

3 Drawing from a Bag
Exercise 3
A bag contains 3 red marbles, 4 blue marbles, and 5 green marbles. What is the probability of drawing a blue marble?
Definition:

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

Total Marbles
3+4+5=12
Blue Marbles
4
Probability
4/12=1/3
Step 1: Find total number of marbles

Total = Red + Blue + Green = 3 + 4 + 5 = 12

Total possible outcomes = 12

Step 2: Identify favorable outcomes

We want to draw a blue marble

Number of blue marbles = 4

Step 3: Apply the probability formula

Probability = 4 ÷ 12 = 4/12 = 1/3

P(blue marble) = 1/3
Final answer:

The probability of drawing a blue marble is 1/3

Applied rules:

• Formula: P(event) = (favorable outcomes)/(total outcomes)

• Simplification: Reduce fractions to lowest terms

• Sample space: All possible outcomes in the experiment

Simple Probability Fundamentals
P(E) = \(\frac{Number\ of\ favorable\ outcomes}{Total\ number\ of\ outcomes}\)
Probability Formula
Certain Event
P = 1
Will definitely occur
Impossible Event
P = 0
Will never occur
Likely Event
P > 0.5
More than 50% chance
Key definitions:

Simple Probability: The probability of a single event occurring

Event: A specific outcome or set of outcomes

Outcome: A possible result of an experiment

Sample Space: The set of all possible outcomes

Favorable Outcome: An outcome that satisfies the event

Complementary Event: The opposite of the desired event

Equally Likely: All outcomes have the same probability

Probability Calculation Process:
  1. Define the experiment: Identify what is happening
  2. List the sample space: Write all possible outcomes
  3. Identify the event: Determine what you're looking for
  4. Count favorable outcomes: Count outcomes that satisfy the event
  5. Apply the formula: Divide favorable by total outcomes
  6. Simplify if possible: Reduce the fraction
Tip 1: Always ensure all outcomes are equally likely before using the formula.
Tip 2: Probability can be expressed as a fraction, decimal, or percent.
Tip 3: The sum of all probabilities in a sample space equals 1.
Tip 4: Complementary events always sum to 1: P(A) + P(not A) = 1.
Common errors: Forgetting to count all outcomes, not simplifying fractions, confusing favorable with total.
Exam preparation: Practice identifying sample spaces, counting outcomes, calculating probabilities.
Solution: Exercises 4 to 5
4 Playing Cards
Exercise 4
A standard deck has 52 cards. What is the probability of drawing a heart or a king?
Definition:

Compound Event: An event that combines multiple outcomes using "or" or "and"

Hearts
13 cards
Kings
4 cards
Overlap
King of Hearts
Total
13+4-1=16
Step 1: Count hearts

There are 13 hearts in a deck

Step 2: Count kings

There are 4 kings in a deck

Step 3: Account for overlap

The King of Hearts is counted in both categories

So we subtract 1 to avoid double counting

Step 4: Apply the formula

Favorable outcomes = 13 + 4 - 1 = 16

Probability = 16 ÷ 52 = 16/52 = 4/13

P(heart or king) = 4/13
Final answer:

The probability of drawing a heart or a king is 4/13

Applied rules:

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

• Avoid double counting: Subtract overlapping outcomes

• Simplification: Reduce fractions to lowest terms

5 Complementary Events
Exercise 5
What is the probability of NOT rolling a 6 on a standard die? Use the complement rule.
Definition:

Complement Rule: P(not A) = 1 - P(A), where A and not A are complementary events

P(Rolling 6)
1/6
P(Not Rolling 6)
1 - 1/6 = 5/6
Step 1: Find P(rolling a 6)

Favorable outcomes = 1 (only rolling a 6)

Total outcomes = 6 (numbers 1-6)

P(rolling a 6) = 1/6

Step 2: Apply the complement rule

P(not rolling a 6) = 1 - P(rolling a 6)

P(not rolling a 6) = 1 - 1/6 = 6/6 - 1/6 = 5/6

Step 3: Verify by direct counting

Outcomes that are NOT 6: {1, 2, 3, 4, 5} = 5 outcomes

Probability = 5/6 ✓

P(not rolling 6) = 5/6
Final answer:

The probability of NOT rolling a 6 is 5/6

Applied rules:

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

• Verification: Can solve by direct counting

• Efficiency: Complement rule saves time when direct counting is difficult

Simple Probability Analysis Summary
P(E) + P(\overline{E}) = 1
Complement Rule
Key definitions:

Simple Probability: The likelihood of a single event occurring

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

Event: A subset of the sample space that we're interested in

Outcome: A single possible result of an experiment

Favorable Outcome: An outcome that satisfies our event

Complementary Event: All outcomes that are NOT in the original event

Equally Likely Outcomes: All outcomes have the same probability of occurring

Complete Probability Analysis:
  1. Experiment identification: Clearly define what is happening
  2. Sample space listing: Enumerate all possible outcomes
  3. Event definition: Specify what you're looking for
  4. Outcome counting: Count favorable and total outcomes
  5. Formula application: Apply probability formula
  6. Result interpretation: Express probability in appropriate form
Tip 1: Always check that all outcomes are equally likely before using the formula.
Tip 2: Use the complement rule when the event has many outcomes to avoid.
Tip 3: Remember that probability ranges from 0 (impossible) to 1 (certain).
Tip 4: When combining events with "or," watch for overlaps that need to be subtracted.
Applications: Used in games, weather forecasting, risk assessment, and decision-making.
Limitations: Assumes equally likely outcomes; doesn't account for real-world complexities.
Essential Formulas:

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

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

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

• Range: 0 ≤ P(E) ≤ 1

Questions & Answers

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

Answer: These are two different approaches to probability:

  • Theoretical probability: Calculated using mathematical reasoning based on the structure of the experiment
  • Experimental probability: Determined by actually performing the experiment multiple times

For example, with a fair coin:
Theoretical: P(heads) = 1/2 (based on structure)
Experimental: Flip 100 times, get 48 heads → P(heads) = 48/100 = 0.48

The Law of Large Numbers states that as trials increase, experimental probability approaches theoretical probability.

Question: Can probability ever be greater than 1 or less than 0? Why?

Answer: No, probability cannot be greater than 1 or less than 0:

  • Probability of 0: Impossible event (never occurs)
  • Probability of 1: Certain event (always occurs)
  • Between 0 and 1: Possible events with varying likelihood

This is because probability is defined as favorable outcomes divided by total outcomes. Since favorable outcomes cannot exceed total outcomes, the fraction cannot be greater than 1. Since favorable outcomes cannot be negative, the fraction cannot be less than 0.

A probability of 0.5 means a 50-50 chance, while values closer to 1 indicate higher likelihood.

Question: When do I use the complement rule instead of calculating directly? What's the advantage?

Answer: Use the complement rule when:

  1. Many favorable outcomes: It's easier to count what you DON'T want
  2. At least one type problems: "At least one" is easier as 1 - "none"
  3. Complex events: When direct counting is complicated

For example, finding P(at least one head in 5 coin flips) is easier as 1 - P(no heads) = 1 - (1/2)⁵ = 1 - 1/32 = 31/32.

The advantage is efficiency - sometimes it's much quicker to count what you don't want than what you do want.

Question: How do I handle compound events with "or" when there might be overlap between the events?

Answer: When events can overlap, use the general addition rule:

Formula: P(A or B) = P(A) + P(B) - P(A and B)

This prevents double counting outcomes that satisfy both events.

For example, drawing a face card OR a heart from a deck:
P(face card) = 12/52
P(heart) = 13/52
P(face card AND heart) = 3/52 (King, Queen, Jack of hearts)
P(face card OR heart) = 12/52 + 13/52 - 3/52 = 22/52

If events are mutually exclusive (no overlap), P(A and B) = 0, so P(A or B) = P(A) + P(B).

Question: What if outcomes aren't equally likely? Can I still use the simple probability formula?

Answer: No, you cannot use the simple probability formula when outcomes are not equally likely:

Simple probability formula: P(E) = (favorable outcomes)/(total outcomes)

This formula only works when all outcomes in the sample space are equally likely.

For example, if you have a weighted die that lands on 6 twice as often as other numbers, you cannot say P(rolling 6) = 1/6. Instead, you'd need experimental probability or other methods to determine the actual probability.

Always verify that outcomes are equally likely before applying the simple probability formula.