Solved Exercises on Probability Models in Grade 8

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

Solution: Exercises 1 to 3
1 Creating a Probability Model
Exercise 1
Create a probability model for rolling a standard six-sided die. List all outcomes and their probabilities.
Definition:

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

Creating a probability model:
  1. Identify the sample space (all possible outcomes)
  2. Verify that outcomes are equally likely
  3. Calculate the probability for each outcome
  4. Verify that all probabilities sum to 1
Sample Space
{1,2,3,4,5,6}
Each Outcome
P = 1/6
Sum
6 × 1/6 = 1
Step 1: Identify the sample space

A standard die has 6 faces numbered 1 through 6

Sample space S = {1, 2, 3, 4, 5, 6}

Step 2: Verify equally likely outcomes

Assuming a fair die, each face has equal chance of appearing

All outcomes are equally likely

Step 3: Calculate individual probabilities

For each outcome: P(outcome) = 1/6

P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1/6

Step 4: Verify the model

Sum of all probabilities = 1/6 + 1/6 + 1/6 + 1/6 + 1/6 + 1/6 = 6/6 = 1

P(n) = 1/6 for n = 1,2,3,4,5,6
Final answer:

Probability model for rolling a die:
P(1) = 1/6, P(2) = 1/6, P(3) = 1/6, P(4) = 1/6, P(5) = 1/6, P(6) = 1/6

Applied rules:

• Equally likely: Each outcome has same probability

• Sum rule: All probabilities must sum to 1

• Sample space: Must include all possible outcomes

2 Validating a Probability Model
Exercise 2
A student claims that when flipping a coin, the probability of heads is 0.6 and tails is 0.5. Is this a valid probability model? Explain.
Definition:

Valid Probability Model: A model where all probabilities are between 0 and 1, and the sum of all probabilities equals 1

Claimed P(Heads)
0.6
Claimed P(Tails)
0.5
Sum
0.6 + 0.5 = 1.1
Step 1: Identify claimed probabilities

P(Heads) = 0.6, P(Tails) = 0.5

Step 2: Check if probabilities sum to 1

Sum = 0.6 + 0.5 = 1.1

Since 1.1 ≠ 1, this violates the sum rule

Step 3: Check probability bounds

Both probabilities are between 0 and 1, so they satisfy the bound rule

Step 4: Conclusion

Since the sum is not 1, this is not a valid probability model

Invalid model: Sum = 1.1 ≠ 1
Final answer:

No, this is not a valid probability model because the probabilities sum to 1.1, not 1. A valid model must have probabilities that sum to exactly 1.

Applied rules:

• Sum rule: All probabilities must sum to 1

• Bound rule: Each probability must be between 0 and 1

• Validation: Check both conditions for validity

3 Using a Probability Model
Exercise 3
A spinner has four equal sections: red, blue, green, and yellow. The probability model is P(red)=0.25, P(blue)=0.25, P(green)=0.25, P(yellow)=0.25. What is the probability of not landing on blue?
Definition:

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

P(Blue)
0.25
P(Not Blue)
1 - 0.25 = 0.75
Alternative
P(Red) + P(Green) + P(Yellow)
Step 1: Identify the probability model

P(Red) = 0.25, P(Blue) = 0.25, P(Green) = 0.25, P(Yellow) = 0.25

All probabilities sum to 1.00 ✓

Step 2: Apply the complement rule

P(Not Blue) = 1 - P(Blue)

P(Not Blue) = 1 - 0.25 = 0.75

Step 3: Alternative method

P(Not Blue) = P(Red) + P(Green) + P(Yellow)

P(Not Blue) = 0.25 + 0.25 + 0.25 = 0.75

Step 4: Verify consistency

Both methods give the same result: 0.75

P(Not Blue) = 0.75
Final answer:

The probability of not landing on blue is 0.75

Applied rules:

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

• Sum rule: P(A) + P(not A) = 1

• Probability model: Use given probabilities to calculate new events

Probability Models Fundamentals
\sum P(outcomes) = 1
Probability Sum Rule
Valid Model
0 ≤ P ≤ 1, ∑P = 1
Must satisfy both conditions
Sample Space
S = {all outcomes}
Complete set of outcomes
Complement
P(not A) = 1 - P(A)
Useful for calculations
Key definitions:

Probability Model: A mathematical representation of a random phenomenon

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

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

Valid Probability Model: A model where all probabilities are between 0 and 1, and sum to 1

Complement Rule: P(not A) = 1 - P(A)

Equally Likely Outcomes: Outcomes that have the same probability

Experimental Probability: Probability based on actual observations

Theoretical Probability: Probability based on mathematical analysis

Creating and Using Probability Models:
  1. Identify experiment: Clearly define what is happening
  2. List outcomes: Enumerate all possible outcomes
  3. Determine probabilities: Assign probabilities to each outcome
  4. Validate model: Check that probabilities sum to 1
  5. Apply model: Use the model to calculate event probabilities
  6. Interpret results: Understand what the probabilities mean
Tip 1: Always verify that your probability model sums to 1.
Tip 2: Use the complement rule when calculating "not" probabilities.
Tip 3: Each probability must be between 0 and 1 inclusive.
Tip 4: Probability models provide a framework for making predictions.
Common errors: Forgetting to check probability sum, assigning probabilities outside [0,1].
Exam preparation: Practice validating models, using complement rule, calculating event probabilities.
Solution: Exercises 4 to 5
4 Constructing a Custom Model
Exercise 4
A bag contains 4 red marbles, 3 blue marbles, and 5 green marbles. Create a probability model for drawing one marble at random.
Definition:

Custom Probability Model: A probability model created from specific experimental conditions or data

Total Marbles
4+3+5=12
P(Red)
4/12 = 1/3
P(Blue)
3/12 = 1/4
P(Green)
5/12
Step 1: Calculate total number of marbles

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

Step 2: Calculate probability for each color

P(Red) = Number of red marbles / Total marbles = 4/12 = 1/3

P(Blue) = Number of blue marbles / Total marbles = 3/12 = 1/4

P(Green) = Number of green marbles / Total marbles = 5/12

Step 3: Verify the model

Sum = 4/12 + 3/12 + 5/12 = 12/12 = 1 ✓

Step 4: Present the complete model

P(Red) = 1/3, P(Blue) = 1/4, P(Green) = 5/12

P(Red)=1/3, P(Blue)=1/4, P(Green)=5/12
Final answer:

The probability model is: P(Red) = 1/3, P(Blue) = 1/4, P(Green) = 5/12

Applied rules:

• Formula: P(Event) = (Number of favorable outcomes)/(Total outcomes)

• Sum rule: All probabilities must sum to 1

• Simplification: Reduce fractions when possible

5 Using a Model for Predictions
Exercise 5
Using the probability model from Exercise 4, what is the probability of drawing a red or green marble? If you draw 100 marbles with replacement, how many would you expect to be blue?
Definition:

Expected Value: The average result over many trials, calculated as number of trials × probability of success

P(Red or Green)
1/3 + 5/12 = 3/4
P(Blue)
1/4
Expected Blue
100 × 1/4 = 25
Step 1: Calculate P(Red or Green)

Since Red and Green are mutually exclusive: P(Red or Green) = P(Red) + P(Green)

P(Red or Green) = 1/3 + 5/12 = 4/12 + 5/12 = 9/12 = 3/4

Step 2: Identify P(Blue)

From Exercise 4: P(Blue) = 1/4

Step 3: Calculate expected number of blue marbles

Expected value = Number of trials × P(Blue)

Expected blue marbles = 100 × 1/4 = 25

Step 4: Interpret the result

Over 100 draws, we expect about 25 blue marbles

P(Red or Green) = 3/4, Expected Blue = 25
Final answer:

The probability of drawing a red or green marble is 3/4. If you draw 100 marbles with replacement, you would expect about 25 to be blue.

Applied rules:

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

• Expected value: Trials × Probability

• Long-term prediction: Probability models predict averages over many trials

Probability Models Analysis Summary
E(X) = n \times P
Expected Value Formula
Key definitions:

Probability Model: A mathematical description of a random phenomenon

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

Valid Model Conditions: All probabilities between 0 and 1, and sum equals 1

Complement Rule: P(not A) = 1 - P(A)

Expected Value: The long-term average result of a random process

Mutually Exclusive Events: Events that cannot occur simultaneously

Probability Distribution: A function that shows the probabilities of all possible outcomes

Complete Probability Model Process:
  1. Define experiment: Clearly state what is happening
  2. Identify sample space: List all possible outcomes
  3. Assign probabilities: Determine probability for each outcome
  4. Validate model: Check that all conditions are met
  5. Apply model: Use the model to calculate event probabilities
  6. Make predictions: Use expected values for long-term results
Tip 1: Always verify that your probability model sums to 1.
Tip 2: Use the complement rule to simplify calculations.
Tip 3: Expected values represent long-term averages, not guaranteed results.
Tip 4: Probability models help make informed decisions under uncertainty.
Applications: Used in games, insurance, finance, weather forecasting, and decision-making.
Limitations: Models are simplifications; real-world conditions may vary.
Essential Rules:

• Probability bounds: 0 ≤ P(outcome) ≤ 1

• Sum rule: ΣP(all outcomes) = 1

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

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

• Expected value: E(X) = n × P(success)

Questions & Answers

Question: What's the difference between a probability model and just calculating probabilities?

Answer: A probability model is the complete framework that describes all possible outcomes and their probabilities, while calculating probabilities is just finding the probability of a specific event:

  • Probability model: Lists P(1), P(2), P(3), P(4), P(5), P(6) for a die
  • Probability calculation: Finding P(even number) = P(2) + P(4) + P(6)

A model is the foundation that allows you to calculate any event probability using the basic probabilities assigned to individual outcomes.

Think of a probability model as the "blueprint" and probability calculations as the "construction" using that blueprint.

Question: Why do probabilities have to sum to 1? What happens if they don't?

Answer: Probabilities sum to 1 because one of the possible outcomes must occur:

Intuitive explanation: When you roll a die, you're certain that one of {1,2,3,4,5,6} will appear. The total certainty is represented by 1.

Mathematical reason: If P(Sample Space) ≠ 1, then there's either missing probability (something that could happen but isn't accounted for) or extra probability (more than certainty).

What happens if they don't sum to 1: The model is invalid and cannot be used for probability calculations. It violates the fundamental axioms of probability theory.

Question: How do I know if a probability model is "fair" or realistic?

Answer: A probability model is "fair" when outcomes are equally likely:

  • Fair coin: P(Heads) = P(Tails) = 0.5
  • Fair die: P(each number) = 1/6
  • Realistic models: Based on actual data or physical properties

To check fairness and realism:
- Are outcomes equally likely based on the physical setup?
- Does the model match experimental data?
- Do all probabilities sum to 1?
- Are probabilities between 0 and 1?

A model is realistic if it accurately reflects the real-world situation it represents.

Question: What's the difference between expected value and actual result? Why don't they always match?

Answer: Expected value is the theoretical average over many trials, while actual results are what happens in a specific instance:

Expected value: Long-term average prediction (e.g., flip coin 100 times → expect 50 heads)

Actual result: What happens in one set of trials (e.g., might get 48 or 53 heads)

Why they don't always match:
- Random variation exists in any finite number of trials
- The Law of Large Numbers says they converge as trials increase
- Expected value is theoretical, actual is empirical

Expected values guide predictions, but actual results may vary due to chance.

Question: Can a probability model have outcomes with probability 0? What about probability 1?

Answer: Yes, probability models can include outcomes with probability 0 or 1:

  • Probability 0: Impossible event (e.g., rolling 7 on a standard die)
  • Probability 1: Certain event (e.g., rolling 1-6 on a standard die)

However, for a standard probability model where we're interested in the possible outcomes:
- All outcomes in the sample space should have probability ≥ 0
- If an outcome has probability 1, then all other outcomes must have probability 0
- Typically, we only include outcomes that are possible (probability > 0)

The important constraint is that all probabilities must be between 0 and 1 inclusive.