Practice Maths

Relative Frequency and Experimental Probability

Key Ideas

Key Terms

Experimental probability
Probability estimated from actual trials: frequency of event ÷ total number of trials. Also called relative frequency.
Relative frequency
The proportion of trials in which an event occurs; used as an estimate of probability from real data.
Frequency
The number of times a particular outcome occurs in an experiment.
Trial
One run of an experiment (e.g. one flip of a coin or one roll of a die).
Theoretical probability
Probability calculated from equally likely outcomes using logic, without needing to run an experiment.
Law of Large Numbers
As the number of trials increases, experimental probability gets closer and closer to the theoretical probability.

Formula

Relative frequency = f ÷ n, where f = frequency of the event and n = total number of trials.

Hot Tip Relative frequencies must sum to 1 (or 100%). If they do not, check for errors in your data or calculation. Always state whether a probability is theoretical or experimental.

Worked Example

Question: A coin is flipped 200 times. It lands heads 114 times. Find the experimental probability of heads and compare it to the theoretical probability.

Step 1 — Experimental probability:
P(heads) = 114/200 = 0.57 = 57%

Step 2 — Theoretical probability:
P(heads) = 1/2 = 0.5 = 50%

Step 3 — Compare:
The experimental result (0.57) is close to but not equal to the theoretical value (0.5). With more trials, the experimental probability would be expected to get closer to 0.5 (law of large numbers).

Theoretical vs Experimental Probability

So far you have learnt about theoretical probability — the probability you calculate from logic and symmetry. But in real life, we often cannot calculate a probability from theory. We don't know if a coin is perfectly fair, or how likely a biased spinner is to land on red. In those cases, we run an experiment and record the actual results.

Experimental probability (also called relative frequency) is calculated from the results of real trials:

Experimental P(event) = frequency of the event ÷ total number of trials

Example: You flip a coin 50 times and get 28 heads. Experimental P(Head) = 28/50 = 0.56. The theoretical probability is 0.5, and your experimental result is close but not exactly the same — that's normal.

The Law of Large Numbers

The law of large numbers is one of the most important ideas in probability: as you increase the number of trials, the experimental probability gets closer and closer to the theoretical probability.

Think of it this way: if you flip a coin just 4 times, you might easily get 3 or even 4 heads by chance. That gives experimental probability of 0.75 or 1.0 — quite far from 0.5. But if you flip it 10 000 times, you would expect to get very close to 5000 heads, giving experimental probability very close to 0.5.

This is why scientists and statisticians use large samples. The more data you collect, the more reliable and accurate your experimental probability becomes.

Comparing Experimental to Theoretical Probability

When you run an experiment and compare your results to the theoretical probabilities, you can draw useful conclusions. If they match closely, the experiment confirms the theory (e.g. the coin appears to be fair). If they differ significantly, one of two things might be true: either the object is biased (not fair), or you haven't done enough trials yet.

For example, suppose you roll a die 60 times and get a "6" only 4 times. Experimental P(6) = 4/60 = 1/15, but theoretical P(6) = 1/6 = 10/60. That seems like a big difference. But with only 60 trials, results can vary quite a bit by chance. You would need to roll many more times to conclude the die is biased.

Simulations

A simulation is an artificial experiment that models a real situation using a random device — like a coin, die, spinner, or random number generator. Simulations allow you to estimate probabilities for situations where real experiments would be too expensive, dangerous, or time-consuming.

Example: What is the probability that a family of 3 children has exactly 2 girls? You could model this by flipping a coin 3 times (H = girl, T = boy), repeating 100 times, and counting how often exactly 2 heads appear. The experimental probability from the simulation would be an estimate of the true theoretical probability (which is 3/8 = 0.375).

Random number generators on calculators or spreadsheets are commonly used in simulations. The more repetitions, the better the estimate.

Recording and Displaying Experimental Data

When running an experiment, keep a tally of outcomes as you go. A tally table is the most organised way to record results. After all trials are complete, calculate the relative frequency for each outcome by dividing by the total number of trials. Check that all relative frequencies add up to 1 (or 100%).

You can then compare your experimental frequencies to the theoretical probabilities in a table or bar chart to see how close your experiment came to the theory.

Key tip: Do not be alarmed if your experimental probability does not exactly match the theoretical probability — some difference is completely normal and expected. What matters is the trend: with more and more trials, the experimental probability should gradually get closer to the theoretical value. If after a large number of trials (say, 500+) your experimental result is still very far from theory, that is evidence that the situation may not be as fair as assumed.

Mastery Practice

  1. A die was rolled 60 times. The results are shown in the table below. Calculate the relative frequency for each outcome. Fluency

    Outcome 1 2 3 4 5 6
    Frequency 8 12 9 11 10 10
    1. Calculate the relative frequency for each of the six outcomes.
    2. What is the theoretical probability of each outcome for a fair die?
    3. Which outcome had the highest relative frequency? Is this surprising?
    4. Do the relative frequencies sum to 1? Verify.

  2. Calculate the experimental probability for each situation. Express as a fraction, decimal, and percentage. Fluency

    1. A coin is tossed 50 times; it lands on heads 28 times. Find P(heads).
    2. A drawing pin is dropped 100 times; it lands point up 63 times. Find P(point up).
    3. A tennis player serves 80 times and makes 52 successful first serves. Find P(successful first serve).
    4. A bag of mixed lollies contains unknown contents. In 40 draws (with replacement), a red lolly was drawn 14 times. Find P(red lolly).
    5. A basketball player attempts 25 free throws and scores 18. Find P(score).

  3. Use probability to calculate the expected frequency of an event. Fluency

    1. A fair die is rolled 120 times. How many times would you expect to roll a 6?
    2. A fair coin is flipped 300 times. How many tails are expected?
    3. P(rain) on any day is 0.3. In June (30 days), how many rainy days are expected?
    4. A spinner has 5 equal sections: 2 red, 2 blue, 1 green. If spun 200 times, how many times is each colour expected?
    5. A bag has 3 red and 7 blue marbles. One marble is drawn and replaced. In 50 draws, how many reds are expected?

  4. Compare experimental and theoretical probability. Understanding

    1. A coin is flipped 10 times: H, H, T, H, H, T, T, H, H, T.
      1. Find the experimental probability of heads.
      2. Compare with the theoretical probability.
      3. If the experiment is repeated with 1000 flips, would you expect the experimental probability to be closer to or further from the theoretical? Explain.
    2. A die is rolled 30 times. Even numbers appeared 18 times.
      1. What is the experimental probability of an even number?
      2. What is the theoretical probability of an even number?
      3. Does the difference suggest the die is biased? Explain your reasoning.

  5. Interpret data and apply the law of large numbers. Understanding

    1. A spinner is spun and the results are recorded. After 10 spins, P(red) = 0.4. After 100 spins, P(red) = 0.32. After 1000 spins, P(red) = 0.253. What does this pattern suggest about the theoretical probability of red? Explain using the law of large numbers.
    2. A factory produces items. In a sample of 200 items, 8 were defective. Estimate P(defective) and use it to predict how many defective items there would be in a batch of 5 000.
    3. In a survey of 50 students, 30 said they preferred science over maths. Estimate P(a student prefers science). Would you trust this estimate more or less if the survey had 500 students? Explain.
    4. A bag contains only red and blue marbles. In 80 draws with replacement, red appeared 32 times. Estimate the number of each colour of marble if there are 10 marbles in total.

  6. Experimental probability problem solving. Problem Solving

    1. A school tracked whether students were late over 40 school days. The student was late on 6 days.
      1. Find the experimental probability of being late.
      2. Estimate the number of days the student would be late in a 200-day school year.
    2. Two students each roll a die 30 times. Student A gets a six 7 times; Student B gets a six 5 times. They combine their results. What is the combined experimental probability of rolling a six, and how does this compare to the theoretical value?
    3. A biased coin has P(heads) = 0.6. If the coin is flipped 500 times, how many more heads than tails would you expect?
    4. Explain in your own words why “I flipped a coin 10 times and got 7 heads, therefore the coin is biased” is not a valid conclusion. What would provide better evidence of bias?
  7. Design and Analyse an Experiment. A spinner has 4 sectors coloured red, blue, green and yellow. You do not know the size of each sector.

    Design and interpret an experiment to estimate the probability of each colour. Problem Solving

    1. After 20 spins the results are: red 4, blue 9, green 4, yellow 3. Calculate the experimental probability of each colour.
    2. After 200 spins the results are: red 38, blue 92, green 41, yellow 29. Recalculate the experimental probability of each colour.
    3. Which set of results (20 spins or 200 spins) gives a more reliable estimate of the true probabilities? Explain using the law of large numbers.
    4. Based on the 200-spin results, estimate the number of times you would expect blue to appear in 500 spins.
  8. Weather Forecasting. A meteorologist records daily rainfall data for a town.

    Use the data below to answer each question. Problem Solving

    MonthDays in monthRainy daysExperimental P(rain)
    January3112 
    February2818 
    March318 
    1. Complete the table by calculating the experimental probability of rain for each month.
    2. Across all three months combined (90 days, 38 rainy days), what is the overall experimental P(rain)?
    3. Use the combined probability to predict how many rainy days to expect in a 365-day year.
    4. Why might the experimental probability differ between months? Is this a limitation of using experimental probability?
  9. Quality Control. A factory tests batches of light bulbs.

    Use experimental probability to solve each quality control problem. Problem Solving

    1. Batch A: 500 bulbs tested, 15 faulty. Batch B: 800 bulbs tested, 32 faulty. Which batch has the higher experimental probability of being faulty?
    2. The factory's target is a fault rate of no more than 2%. Do either batch A or batch B meet this target?
    3. Using batch A’s fault rate, estimate the number of faulty bulbs expected in a production run of 10 000 bulbs.
    4. The factory improves its process. In the next 1000 bulbs, only 8 are faulty. Has the improvement been effective? Justify using experimental probability.
  10. Critical Thinking. Evaluate the following experimental probability scenarios.

    Evaluate and critique each use of experimental probability. Problem Solving

    1. A market researcher surveys 5 people and finds 4 prefer Brand X. She concludes that 80% of all consumers prefer Brand X. Identify two problems with this conclusion.
    2. A die is rolled 600 times. The frequency of each outcome is: 1→95, 2→102, 3→99, 4→101, 5→98, 6→105. Is the die likely to be fair? Use relative frequencies to justify your answer.
    3. A basketball player makes 7 out of her first 10 free throws in a new season. Her coach says “Your true free throw probability is 70%.” Evaluate this claim. What additional information would make the estimate more reliable?
    4. Two students each estimate P(a randomly chosen student owns a pet). Student A surveys 8 friends and gets 0.625; Student B surveys 80 students and gets 0.55. Whose estimate should be trusted more? Explain.
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