🏠 Home › 📚 CSEC Mathematics › 📊 Module 3: Statistics

Experimental and Theoretical Probability

CSEC Mathematics: Predicting the Future

Essential Understanding: Probability measures the likelihood that an event will occur. Experimental probability comes from real trials, while theoretical probability is based on mathematical reasoning. As trials increase, experimental probability approaches theoretical probability - this is the Law of Large Numbers, a fundamental principle in statistics.

🔑 Key Skill: Calculating Probability

📈 Exam Focus: Law of Large Numbers

🎯 Problem Solving: Complementary Events

Understanding Probability

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Experimental Probability

Definition: Probability based on actual observations or trials. Also called empirical probability.

Formula:

                $$ P(E) = \frac{\text{Number of times event occurs}}{\text{Total number of trials}} $$
            

Example: Tossing a coin 100 times and getting 45 heads.

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Theoretical Probability

Definition: Probability based on mathematical reasoning about equally likely outcomes.

Formula:

                $$ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} $$
            

Example: Fair coin has P(heads) = 1/2.

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Law of Large Numbers

Definition: As the number of trials increases, experimental probability approaches theoretical probability.

Implications:

More trials = more reliable results
Short-term variations smooth out over time
Theoretical value is the "long-run" expectation

Key Probability Rules

Essential Probability Properties

Probability always satisfies these rules:

Probability Range

$$ 0 \leq P(E) \leq 1 $$

0 = impossible, 1 = certain

Complementary Events

$$ P(E) + P(E') = 1 $$

E' = "not E"

Addition Rule (Mutually Exclusive)

$$ P(A \text{ or } B) = P(A) + P(B) $$

When A and B cannot happen together

📝 Worked Example: Calculating Theoretical Probability

Problem 1: A fair die is rolled once. Find the probability of:

Getting an even number
Getting a number greater than 4

Solution:

Sample Space: {1, 2, 3, 4, 5, 6}, Total = 6 outcomes

(a) Even number: Even numbers = {2, 4, 6}, so 3 outcomes

$$ P(\text{even}) = \frac{3}{6} = \frac{1}{2} = 0.5 $$

(b) Number greater than 4: Numbers = {5, 6}, so 2 outcomes

$$ P(>4) = \frac{2}{6} = \frac{1}{3} \approx 0.333 $$

📝 Worked Example: Complementary Events

Problem: The probability of passing a test is 0.7. Find the probability of failing.

Solution:

Using complementary rule:

                $$ P(\text{fail}) = 1 - P(\text{pass}) $$
                $$ P(\text{fail}) = 1 - 0.7 = 0.3 $$
            

Answer: Probability of failing = 0.3 (or 30%)

Interactive Dice Simulation

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Dice Rolling Experiment

Objective: Observe how experimental probability approaches theoretical probability (1/6 = 0.167) as the number of trials increases.

Total Rolls

Sixes Rolled

Experimental P(6)

Theoretical P(6)

0.167

Addition Rule for Mutually Exclusive Events

Addition Rule

When two events cannot occur at the same time (mutually exclusive):

$$ P(A \cup B) = P(A) + P(B) $$

Or: P(A or B) = P(A) + P(B)

📝 Worked Example: Addition Rule

Problem: From a standard deck of 52 cards, what is the probability of drawing a heart OR a diamond?

Solution:

These are mutually exclusive events (can't be both heart AND diamond)

P(heart) = 13/52 (13 hearts in deck)

P(diamond) = 13/52 (13 diamonds in deck)

Using addition rule:

                $$ P(\text{heart or diamond}) = \frac{13}{52} + \frac{13}{52} = \frac{26}{52} = \frac{1}{2} $$
            

Answer: 0.5 or 50%

Probability Comparison Chart

Making Inferences from Probability

Using Probability for Predictions

Calculate Expected Frequency: Expected frequency = Probability × Number of trials

Make Predictions: Use this to predict how many times an event will occur

Compare with Actual: The difference shows natural variation expected in experiments

📝 Worked Example: Making Predictions

Problem: A spinner has 8 equal sections numbered 1-8. If spun 400 times, how many times would you expect to land on a prime number?

Solution:

Prime numbers between 1-8: 2, 3, 5, 7 (4 primes)

P(prime) = 4/8 = 0.5

Expected frequency:

$$ \text{Expected} = 0.5 \times 400 = 200 \text{ times} $$

Prediction: The spinner should land on a prime number approximately 200 times out of 400.

Key Examination Insights

Common Mistakes

Confusing experimental and theoretical probability formulas
Forgetting that probability is always between 0 and 1
Using addition rule when events are NOT mutually exclusive
Not simplifying fractions in final answers

Success Strategies

Always identify sample space first
Check if events are mutually exclusive before adding
Use complementary rule (1 - P) when "at least" or "not" is in the question
Express answers as fractions, decimals, or percentages

CSEC Practice Arena

Test Your Understanding

A bag contains 5 red, 3 blue, and 2 green balls. What is the probability of drawing a red ball?

1/2

5/10 = 1/2

3/10

2/10

Solution: Total balls = 5 + 3 + 2 = 10. Red balls = 5. P(red) = 5/10 = 1/2.

What is the complement of rolling a 6 on a fair die?

Rolling a 1

Rolling an even number

Not rolling a 6

Rolling a 5

Explanation: The complement of event E (rolling a 6) is "not E" (not rolling a 6). This includes rolling 1, 2, 3, 4, or 5.

What does the Law of Large Numbers state?

Probability can never exceed 1

All events are equally likely

Experimental probability approaches theoretical probability as trials increase

The sum of all probabilities equals 1

Explanation: The Law of Large Numbers states that as more trials are conducted, the experimental (observed) probability will get closer and closer to the theoretical (expected) probability.

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CSEC Examination Mastery Tip

Probability Questions: When solving CSEC probability questions:

Always write the probability as a fraction in simplest form
For "at least" or "not" questions, use the complementary rule
For "or" questions, check if events are mutually exclusive first
Show your working: sample space, favorable outcomes, calculation
Remember: P(certain) = 1, P(impossible) = 0