Solved Exercises on Theoretical Probability in Grade 8

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

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

Theoretical Probability: The probability of an event based on the possible outcomes, calculated as favorable outcomes divided by total possible outcomes

Theoretical probability method:
  1. Identify the sample space (all possible outcomes)
  2. Identify favorable outcomes (outcomes that satisfy the event)
  3. Apply the theoretical probability formula
  4. Simplify the fraction if possible
Sample Space
{1,2,3,4,5,6}
Prime Numbers
{2,3,5}
Probability
3/6 = 1/2
Step 1: Identify the sample space

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

Total possible outcomes = 6

Step 2: Identify prime numbers

Prime numbers are numbers greater than 1 that have only two factors: 1 and themselves

Prime numbers in {1,2,3,4,5,6}: {2, 3, 5}

Favorable outcomes = 3

Step 3: Apply the probability formula

Theoretical Probability = (Favorable outcomes)/(Total outcomes)

Theoretical Probability = 3/6 = 1/2

P(prime number) = 1/2
Final answer:

The theoretical probability of rolling a prime number is 1/2

Applied rules:

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

• Prime numbers: 2, 3, 5, 7, 11, ...

• Equally likely: Each outcome has equal probability

2 Flipping Two Coins
Exercise 2
What is the theoretical probability of getting exactly one head when flipping two coins?
Definition:

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

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

When flipping two coins, all possible outcomes are:

{HH, HT, TH, TT} where H=heads, T=tails

Total outcomes = 4

Step 2: Identify favorable outcomes

Outcomes with exactly one head: {HT, TH}

Favorable outcomes = 2

Step 3: Calculate the probability

Theoretical Probability = 2/4 = 1/2

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

The theoretical probability of getting exactly one head is 1/2

Applied rules:

• Sample space: List all possible outcomes systematically

• Order matters: HT and TH are different outcomes

• Counting: Ensure no outcomes are missed or duplicated

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

Equally Likely Outcomes: Each outcome has the same probability of occurring

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

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

Total possible outcomes = 12

Step 2: Identify favorable outcomes

Number of blue marbles = 3

Favorable outcomes = 3

Step 3: Calculate the probability

Theoretical Probability = 3/12 = 1/4

P(blue marble) = 1/4
Final answer:

The theoretical probability of drawing a blue marble is 1/4

Applied rules:

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

• Simplification: Reduce fractions to lowest terms

• Equally likely: Each marble has equal chance of being drawn

Theoretical Probability Fundamentals
P(E) = \(\frac{Number\ of\ favorable\ outcomes}{Total\ number\ of\ possible\ outcomes}\)
Theoretical Probability Formula
Sample Space
S = {all outcomes}
Set of all possibilities
Favorable Outcomes
Subset of S
Outcomes that satisfy E
Equally Likely
Each outcome = same P
Required for formula
Key definitions:

Theoretical Probability: The probability of an event based on mathematical analysis of possible outcomes

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

Event: A specific outcome or set of outcomes from the sample space

Favorable Outcomes: Outcomes that satisfy the event we're interested in

Equally Likely Outcomes: Outcomes that have the same probability of occurring

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

Impossible Event: An event with probability 0

Certain Event: An event with probability 1

Theoretical Probability Process:
  1. Define the experiment: Clearly state what is happening
  2. Identify the sample space: List all possible outcomes
  3. Define the event: Specify 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: Reduce the fraction to lowest terms
Tip 1: Always verify that outcomes are equally likely before using the formula.
Tip 2: Organize sample spaces systematically to avoid missing outcomes.
Tip 3: Use tree diagrams or tables for complex sample spaces.
Tip 4: The sum of all individual outcome probabilities equals 1.
Common errors: Missing outcomes in sample space, counting errors, assuming unequal probabilities.
Exam preparation: Practice identifying sample spaces, counting outcomes, and applying formulas.
Solution: Exercises 4 to 5
4 Drawing Cards
Exercise 4
A standard deck of 52 cards is shuffled. What is the theoretical probability of drawing a face card (Jack, Queen, or King)?
Definition:

Face Cards: Jacks, Queens, and Kings in a deck of cards

Total Cards
52
Face Cards
12
Probability
12/52 = 3/13
Step 1: Identify total possible outcomes

A standard deck has 52 cards

Total outcomes = 52

Step 2: Count face cards

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

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

Favorable outcomes = 12

Step 3: Calculate the probability

Theoretical Probability = 12/52 = 3/13

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

The theoretical probability of drawing a face card is 3/13

Applied rules:

• Card counting: 4 suits, 3 face cards per suit

• Simplification: 12/52 reduces to 3/13

• Standard deck: 52 cards total

5 Complementary Events
Exercise 5
Using the complement rule, find the theoretical probability of NOT rolling a multiple of 3 on a standard die.
Definition:

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

Multiples of 3
{3,6}
P(multiple of 3)
2/6 = 1/3
P(not multiple of 3)
1 - 1/3 = 2/3
Step 1: Find P(rolling a multiple of 3)

On a standard die {1,2,3,4,5,6}, multiples of 3 are {3,6}

P(multiple of 3) = 2/6 = 1/3

Step 2: Apply the complement rule

P(not multiple of 3) = 1 - P(multiple of 3)

P(not multiple of 3) = 1 - 1/3 = 3/3 - 1/3 = 2/3

Step 3: Verify by direct calculation

Numbers that are NOT multiples of 3: {1,2,4,5} = 4 outcomes

P(not multiple of 3) = 4/6 = 2/3 ✓

P(not multiple of 3) = 2/3
Final answer:

The theoretical probability of NOT rolling a multiple of 3 is 2/3

Applied rules:

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

• Verification: Direct calculation confirms the result

• Efficiency: Complement rule can simplify calculations

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

Theoretical Probability: The probability of an event based on mathematical analysis

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

Event: A subset of the sample space representing desired outcomes

Favorable Outcomes: Outcomes that satisfy the event criteria

Equally Likely Outcomes: Outcomes with identical probability

Complementary Event: The event consisting of all outcomes not in the original event

Mutually Exclusive Events: Events that cannot occur simultaneously

Complete Theoretical Probability Analysis:
  1. Experiment definition: Clearly describe the process
  2. Sample space identification: List all possible outcomes systematically
  3. Event specification: Define the specific outcome(s) of interest
  4. Outcome counting: Count favorable and total outcomes
  5. Formula application: Apply theoretical probability formula
  6. Verification: Check reasonableness and calculations
Tip 1: Always ensure outcomes are equally likely before applying the formula.
Tip 2: Use the complement rule when the event has many favorable outcomes.
Tip 3: Organize complex sample spaces using systematic methods.
Tip 4: Theoretical probability ranges from 0 (impossible) to 1 (certain).
Applications: Games, risk assessment, decision-making, and statistical modeling.
Limitations: Assumes equally likely outcomes; may not reflect real-world complexities.
Essential Formulas:

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

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

• Range: 0 ≤ P(E) ≤ 1

• Sum of probabilities: ΣP(all outcomes) = 1

Questions & Answers

Question: How do I know if outcomes are equally likely, and what happens if they're not?

Answer: Outcomes are equally likely when each has the same probability of occurring:

  • Standard die: Each face has equal probability (1/6)
  • Fair coin: Heads and tails equally likely (1/2 each)
  • Shuffled deck: Each card equally likely to be drawn (1/52)

If outcomes are not equally likely, the basic theoretical probability formula doesn't apply. For example, if a die is weighted to favor the number 6, then P(rolling 6) ≠ 1/6. In such cases, you'd need experimental probability or other specialized methods to determine probabilities.

Always verify equal likelihood before using the standard formula.

Question: What's the difference between theoretical probability and experimental probability, and when do I use each?

Answer: These are two different approaches to probability:

  • Theoretical probability: Calculated using mathematical analysis based on possible outcomes
  • Experimental probability: Determined by actually performing the experiment multiple times

Use theoretical probability when outcomes are equally likely and the system is well understood. Use experimental probability when:
- Theoretical probability is unknown or complex
- You want to verify theoretical calculations
- Real-world conditions may affect outcomes
- The system doesn't follow ideal mathematical models

The Law of Large Numbers states that experimental probability approaches theoretical probability as the number of trials increases.

Question: How do I organize sample spaces for complex experiments with multiple stages?

Answer: Use systematic methods to organize complex sample spaces:

  1. Tree diagrams: Branch out each stage of the experiment
  2. Tables: For experiments with two variables (like rolling two dice)
  3. Systematic listing: Use logical order (alphabetical, numerical)
  4. Counting principles: Multiply possibilities at each stage

For example, when rolling two dice, you can list all 36 outcomes systematically:
(1,1), (1,2), ..., (1,6), (2,1), (2,2), ..., (6,6)

The key is to be methodical and ensure no outcomes are missed or duplicated.

Question: When should I use the complement rule instead of calculating directly?

Answer: Use the complement rule when:

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

For example, finding P(at least one head in 5 coin flips) is easier as 1 - P(no heads) = 1 - (1/2)⁵ = 31/32, rather than counting all possibilities with 1, 2, 3, 4, or 5 heads.

The complement rule is especially helpful when the event has many favorable outcomes, making direct calculation tedious.

Question: What if the theoretical probability seems too abstract for a real-world situation? How do I know it's valid?

Answer: Theoretical probability is a mathematical model that makes certain assumptions:

Validity depends on assumptions:
- Fair coin (equally weighted)
- Standard die (perfect cube, uniform density)
- Well-shuffled deck (random order)

When theoretical probability may not match reality:
- Physical imperfections (worn dice, bent coins)
- Environmental factors (wind, temperature)
- Human bias (non-random shuffling)

Theoretical probability provides a baseline expectation. When real-world results differ significantly, it may indicate that the assumptions don't hold, prompting investigation of external factors. For practical purposes, theoretical probability serves as a useful model for decision-making.