Solved Exercises on Experimental Probability in Grade 7

Master experimental probability: calculating likelihood based on trial results, comparing with theoretical probability, and interpreting data through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Coin Tossing Experiment
Exercise 1
Sarah flipped a coin 50 times and got heads 28 times. What is the experimental probability of getting heads? How does this compare to the theoretical probability?
Definition:

Experimental Probability: The probability of an event based on actual experimental results.

Formula: P(E) = (Number of favorable outcomes) ÷ (Total number of trials)

Theoretical Probability: The probability calculated using mathematical principles.

Experimental Probability Method:
  1. Count the number of times the event occurred
  2. Count the total number of trials
  3. Apply the formula: P(E) = (occurrences) ÷ (trials)
  4. Compare with theoretical probability if needed
  5. Express as fraction, decimal, or percentage
Heads Occurred
28 times
Total Trials
50
Exp. Prob.
28/50 = 14/25
Theo. Prob.
1/2
Step 1: Identify the event and outcomes

Event: Getting heads

Number of times heads occurred: 28

Total number of trials: 50

Step 2: Apply experimental probability formula

P(Heads) = Number of heads ÷ Total trials

P(Heads) = 28 ÷ 50 = 28/50

Step 3: Simplify the fraction

28/50 = (28÷2)/(50÷2) = 14/25

Step 4: Compare with theoretical probability

Theoretical probability of heads = 1/2 = 0.5

Experimental probability of heads = 14/25 = 0.56

The experimental probability is close to the theoretical probability.

Exp. P(Heads) = 14/25 or 56%
Final Answer:

The experimental probability of getting heads is 14/25 or 56%. This is close to the theoretical probability of 1/2 or 50%.

Applied Rules:

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

• Comparison: Experimental probability approaches theoretical as trials increase

• Simplification: Reduce fractions to lowest terms

2 Rolling a Die Experiment
Exercise 2
A student rolled a die 60 times and recorded the following results: 1 appeared 12 times, 2 appeared 8 times, 3 appeared 10 times, 4 appeared 9 times, 5 appeared 11 times, and 6 appeared 10 times. What is the experimental probability of rolling a number greater than 4?
Definition:

Rolling a Die: A standard die has 6 faces numbered 1-6.

Numbers Greater Than 4: On a die, these are 5 and 6.

Rolls of 5
11 times
Rolls of 6
10 times
Total Trials
60
Exp. Prob.
21/60 = 7/20
Step 1: Identify the event

Event: Rolling a number greater than 4

This includes rolling a 5 or a 6

Step 2: Count favorable outcomes

Number of times 5 appeared: 11

Number of times 6 appeared: 10

Total favorable outcomes: 11 + 10 = 21

Step 3: Count total trials

Total number of rolls: 60

Step 4: Calculate experimental probability

P(Number > 4) = Favorable outcomes ÷ Total trials

P(Number > 4) = 21 ÷ 60 = 21/60

Step 5: Simplify the fraction

21/60 = (21÷3)/(60÷3) = 7/20

P(Number > 4) = 7/20 or 35%
Final Answer:

The experimental probability of rolling a number greater than 4 is 7/20 or 35%.

Applied Rules:

• Experimental Probability: P(E) = (favorable) ÷ (trials)

• Compound Event: For "or" conditions, add favorable outcomes

• Fraction Simplification: Reduce to lowest terms

3 Comparing Theoretical and Experimental
Exercise 3
A spinner has 4 equal sections colored red, blue, green, and yellow. The theoretical probability of landing on red is 1/4. In an experiment, the spinner was spun 80 times and landed on red 22 times. Calculate the experimental probability and compare it to the theoretical probability.
Definition:

Theoretical Probability: Probability calculated using mathematical principles.

Experimental Probability: Probability based on actual experimental results.

Red Occurred
22 times
Total Spins
80
Exp. Prob.
22/80 = 11/40
Theo. Prob.
1/4
Step 1: Calculate theoretical probability

Spinner has 4 equal sections

Theoretical P(Red) = 1/4 = 0.25

Step 2: Calculate experimental probability

Red occurred: 22 times

Total spins: 80

Experimental P(Red) = 22/80 = 11/40 = 0.275

Step 3: Compare the probabilities

Theoretical: 1/4 = 0.25

Experimental: 11/40 = 0.275

Difference: 0.275 - 0.25 = 0.025

Step 4: Analyze the results

The experimental probability (0.275) is slightly higher than theoretical (0.25)

With more trials, the experimental probability should approach the theoretical probability.

Theo. P(Red) = 1/4 = 25%
Exp. P(Red) = 11/40 = 27.5%
Final Answer:

Theoretical probability: 1/4 or 25%. Experimental probability: 11/40 or 27.5%. The experimental probability is slightly higher but close to the theoretical probability.

Applied Rules:

• Law of Large Numbers: As trials increase, experimental approaches theoretical

• Comparison: Calculate difference between experimental and theoretical

• Validation: Small differences are expected in experiments

Rules and methods, laws,...
\(P_{exp}(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of trials}}\)
Experimental Probability Formula
Experimental vs Theoretical
Experimental: Actual trials
Based on observations
Law of Large Numbers
As trials increase, exp → theoretical
Approaches true probability
Relative Frequency
Same as experimental probability
Ratio of occurrences
Key Definitions:

Experimental Probability: The probability of an event based on actual experimental results, calculated as favorable outcomes divided by total trials

Theoretical Probability: The probability calculated using mathematical principles, based on possible outcomes

Trials: The number of times an experiment is repeated

Relative Frequency: Another term for experimental probability

Law of Large Numbers: As the number of trials increases, experimental probability approaches theoretical probability

Favorable Outcomes: The outcomes that satisfy the event condition

Empirical Probability: Another term for experimental probability

Complete Methodology:
  1. Perform Experiment: Conduct the trials and record results
  2. Count Outcomes: Count how many times the event occurred
  3. Count Trials: Count the total number of trials conducted
  4. Apply Formula: Calculate P(E) = (favorable outcomes) ÷ (total trials)
  5. Simplify: Reduce fraction to lowest terms if possible
  6. Compare: Compare with theoretical probability if known
Tip 1: Always record results carefully during experiments.
Tip 2: More trials lead to more accurate experimental probability.
Tip 3: Experimental probability may differ from theoretical due to randomness.
Tip 4: Use the law of large numbers to predict convergence.
Tip 5: Experimental probability is always based on actual data, not assumptions.
Common Errors: Miscounting outcomes, confusing experimental with theoretical, not simplifying fractions, expecting exact match with theory.
Exam Preparation: Practice with various experiments, memorize the formula, understand the relationship with theoretical probability, know how to interpret differences.
Solution: Exercises 4 to 5
4 Drawing Marbles
Exercise 4
A bag contains 5 red marbles, 3 blue marbles, and 2 green marbles. A marble is drawn at random, its color recorded, and then returned to the bag. This process is repeated 100 times. The results were: 45 red, 32 blue, and 23 green. Calculate the experimental probability for each color and compare with the theoretical probability.
Definition:

With Replacement: The marble is returned to the bag after each draw, maintaining equal probabilities.

Without Replacement: The marble is not returned, changing probabilities for subsequent draws.

Red Exp.
45/100 = 9/20
Blue Exp.
32/100 = 8/25
Green Exp.
23/100
Step 1: Calculate theoretical probabilities

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

Theoretical P(Red) = 5/10 = 1/2

Theoretical P(Blue) = 3/10

Theoretical P(Green) = 2/10 = 1/5

Step 2: Calculate experimental probabilities

Total trials: 100

Experimental P(Red) = 45/100 = 9/20

Experimental P(Blue) = 32/100 = 8/25

Experimental P(Green) = 23/100

Step 3: Compare red probabilities

Theoretical: 1/2 = 50%

Experimental: 9/20 = 45%

Difference: 5%

Step 4: Compare blue probabilities

Theoretical: 3/10 = 30%

Experimental: 8/25 = 32%

Difference: 2%

Step 5: Compare green probabilities

Theoretical: 1/5 = 20%

Experimental: 23/100 = 23%

Difference: 3%

Step 6: Analyze results

Experimental probabilities are close to theoretical ones

Small differences are expected in experiments

Red: Theo=50%, Exp=45%
Blue: Theo=30%, Exp=32%
Green: Theo=20%, Exp=23%
Final Answer:

Experimental probabilities: Red=45%, Blue=32%, Green=23%. These are close to theoretical probabilities (Red=50%, Blue=30%, Green=20%), showing the experimental results align well with theory.

Applied Rules:

• With Replacement: Probabilities remain constant for each trial

• Multiple Events: Calculate probability for each outcome separately

• Comparison: Small differences are normal in experimental probability

5 Real-World Application
Exercise 5
A basketball player attempted 200 free throws during practice and made 165 of them. What is the experimental probability that the player will make a free throw? If the player attempts 40 more free throws in a game, how many would you expect them to make based on the experimental probability?
Definition:

Real-World Application: Using experimental probability to predict future outcomes.

Expected Value: The number of successes predicted based on probability.

Made Throws
165
Total Attempts
200
Exp. Prob.
165/200 = 33/40
Predicted Makes
33
Step 1: Calculate experimental probability

Free throws made: 165

Total attempts: 200

Experimental P(Make) = 165/200 = 33/40 = 0.825

Step 2: Express as percentage

33/40 = 0.825 = 82.5%

Step 3: Calculate expected value for game

Game attempts: 40

Expected makes = P(Make) × Game attempts

Expected makes = (33/40) × 40 = 33

Step 4: Verify the calculation

33/40 × 40 = 33 ✓

Step 5: Interpret the result

Based on past performance, we expect 33 successful free throws out of 40 attempts

Exp. P(Make) = 33/40 = 82.5%
Expected makes = 33
Final Answer:

The experimental probability of making a free throw is 33/40 or 82.5%. Based on this probability, the player is expected to make 33 out of 40 free throws in the game.

Applied Rules:

• Expected Value: Expected = P(event) × Number of trials

• Real-World Prediction: Use past performance to predict future outcomes

• Probability Application: Apply experimental probability to new situations

Detailed Summary: Experimental Probability Fundamentals
\(P_{exp}(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of trials}}\)
Experimental Probability Formula
Key definitions:

Experimental Probability: The probability of an event based on actual experimental results, calculated as the ratio of favorable outcomes to total trials

Theoretical Probability: The probability calculated using mathematical principles and possible outcomes

Trials: The number of times an experiment is repeated

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

Relative Frequency: Another term for experimental probability

Law of Large Numbers: As the number of trials increases, experimental probability approaches theoretical probability

Empirical Probability: Another term for experimental probability

Complete methodology:
  1. Conduct Experiment: Perform the trials and record all outcomes
  2. Count Favorable: Count how many times the event occurred
  3. Count Total: Count the total number of trials conducted
  4. Calculate: Apply P(E) = (favorable) ÷ (trials)
  5. Simplify: Reduce fraction to lowest terms if possible
  6. Compare: Compare with theoretical probability if known
  7. Interpret: Understand what the experimental probability means
Tip 1: Always record experimental results carefully to ensure accuracy.
Tip 2: More trials generally lead to more accurate experimental probabilities.
Tip 3: Small differences between experimental and theoretical are normal.
Tip 4: Use experimental probability to make predictions about future events.
Tip 5: Experimental probability is always based on actual data, not assumptions.
Common errors: Miscounting outcomes, confusing with theoretical probability, not simplifying fractions, expecting exact match with theory, not considering sample size effects.
Exam preparation: Practice with various experiments, memorize the formula, understand the relationship with theoretical probability, know how to interpret differences, practice prediction problems.
Formulas to know by heart:

• Experimental Probability: P(E) = (Number of favorable outcomes) ÷ (Total number of trials)

• Expected Value: Expected = P(E) × Number of trials

• Law of Large Numbers: As trials → ∞, experimental → theoretical

• Probability Range: 0 ≤ P(E) ≤ 1

• Relative Frequency: Same as experimental probability

Exercise with Visualization: Probability Convergence
Exercise 6: Experimental vs Theoretical
Simulate coin flipping experiment and observe how experimental probability converges to theoretical probability:
Trial 1: 10 flips, 6 heads
Trial 2: 50 flips, 28 heads
Trial 3: 100 flips, 52 heads
Trial 4: 500 flips, 248 heads

Analysis: The visualization shows how experimental probability approaches theoretical probability as trials increase.

  • Trial 1: 6/10 = 60% (far from 50%)
  • Trial 2: 28/50 = 56% (closer to 50%)
  • Trial 3: 52/100 = 52% (very close to 50%)
  • Trial 4: 248/500 = 49.6% (very close to 50%)

Questions & Answers

Question: Why is experimental probability different from theoretical probability? Shouldn't they be the same?

Answer: Experimental and theoretical probability are different because:

  • Theoretical Probability: Based on mathematical principles and all possible outcomes
  • Experimental Probability: Based on actual results from a finite number of trials

They don't have to be the same because of randomness in experiments. However, according to the Law of Large Numbers, as the number of trials increases, experimental probability gets closer to theoretical probability.

Example: Theoretical probability of flipping heads = 1/2 = 50%

  • After 10 flips: might get 6 heads (60%) - different
  • After 1000 flips: might get 502 heads (50.2%) - much closer

Small differences in experiments are completely normal and expected!

Question: How many trials do I need to perform to get accurate experimental probability?

Answer: There's no exact number, but generally:

  • More trials = More accurate results
  • 10-20 trials: Rough approximation
  • 50-100 trials: Better approximation
  • 1000+ trials: Very close to theoretical probability

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

For school experiments, 50-100 trials usually give reasonable results. For more precision, conduct more trials.

Remember: Some variation is always expected due to randomness!

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

Answer: No, experimental probability cannot be greater than 1 or less than 0. Here's why:

Experimental probability is calculated as:

P(E) = (Number of favorable outcomes) ÷ (Total number of trials)

Since the number of favorable outcomes cannot exceed the total number of trials, and both numbers are non-negative:

  • Minimum value: 0 ÷ n = 0 (when no favorable outcomes)
  • Maximum value: n ÷ n = 1 (when all trials are favorable)

If you calculate an experimental probability outside the 0-1 range, you made an error in counting:

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

Always verify that your experimental probability is between 0 and 1!

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

Answer: Here are verification methods:

  1. Range Check: Ensure probability is between 0 and 1
  2. Count Verification: Double-check your counts of favorable outcomes and total trials
  3. Reasonableness Check: Does the result make sense given your data?
  4. Decimal Conversion: Convert to decimal and verify it's reasonable
  5. Percentage Check: Convert to percentage and see if it's realistic

Example: If you got 28 heads out of 50 flips:

  • Calculation: 28/50 = 0.56 = 56%
  • Range: 0.56 is between 0 and 1 ✓
  • Reasonable: 56% is close to expected 50% ✓

Always verify your arithmetic and ensure your answer reflects the actual experimental data!

Question: When would I use experimental probability instead of theoretical probability?

Answer: Use experimental probability when:

  • Real-world situations: Sports performance, quality control, weather patterns
  • Unknown theoretical probability: When mathematical model is complex or unknown
  • Testing fairness: Checking if a coin or die is fair
  • Historical data: Predicting based on past performance
  • Complex systems: Situations where theoretical calculation is difficult

Examples:

  • Basketball player's free throw percentage (based on past performance)
  • Machine defect rate (based on inspection data)
  • Weather patterns (based on historical records)

Theoretical probability is ideal when all outcomes are equally likely and known. Experimental probability is more practical when dealing with real-world data!