Expected vs Observed Frequency
Section: Probability | Syllabus: Cambridge Lower Secondary Checkpoint Mathematics (0862)
Expected Frequency
Expected frequency = probability of an event × number of trials.
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Question: A fair die is rolled 60 times. Find the expected frequency of rolling a 4.
- Step 1: P(4) = 1/6
- Step 2: Expected frequency = 1/6 × 60
- Answer: 10
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Question: A spinner has P(red) = 0.3. It is spun 200 times. Find the expected frequency of red.
- Step 1: Expected frequency = 0.3 × 200
- Answer: 60
Observed Frequency and Relative Frequency
Observed frequency is the actual count recorded from doing an experiment. Relative frequency = observed frequency ÷ number of trials - an estimate of the true probability based on experimental data.
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Question: A coin is flipped 50 times and lands on heads 28 times. Find the relative frequency of heads.
- Step 1: Relative frequency = 28 ÷ 50
- Answer: 0.56
Combining Relative Frequency from Multiple Experiments
When two experiments test the same thing, combine them by adding the observed counts and adding the numbers of trials, THEN dividing - not by averaging the two relative frequencies. This correctly gives more weight to the larger experiment.
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Question: Student A spins a spinner 40 times and gets red 9 times. Student B spins the same spinner 60 times and gets red 17 times. Find the combined relative frequency of red.
- Step 1: Total red = 9 + 17 = 26. Total spins = 40 + 60 = 100
- Step 2: Combined relative frequency = 26 ÷ 100
- Answer: 0.26
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Question: Explain why averaging the two relative frequencies, (9/40 + 17/60) ÷ 2, gives a less reliable answer than the pooled method above.
- Answer: Averaging treats both experiments as equally important even though Student B collected more data (60 trials vs 40). Pooling the actual counts correctly gives the larger, more reliable experiment more influence over the combined result.
Comparing Expected and Observed Frequency
Differences between expected and observed frequency can arise from natural random variation (chance), a small number of trials, or the device/experiment actually being biased.
Face 6's observed frequency (18) is notably higher than its expected value (10)
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Question: A fair die is rolled 60 times (expected frequency of each number = 10). The observed frequency of rolling a 6 is 18. Comment on this result.
- 18 is notably higher than the expected 10.
- Answer: This could be due to chance (especially with a limited number of trials), but such a large difference might also suggest the die is biased toward 6 - more trials would help decide which is more likely.
The Law of Large Numbers
As the number of trials increases, relative frequency tends to get closer to the true theoretical probability. Small numbers of trials can show big swings from the expected value purely by chance.
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Question: A coin gives 7 heads in 10 flips (relative frequency 0.7), then 512 heads in 1000 flips (relative frequency 0.512). Which result is likely a better estimate of the true probability of heads?
- Answer: The 1000-flip result (0.512) - larger numbers of trials tend to smooth out random variation and get closer to the true probability (0.5 for a fair coin)
As rolls increase, relative frequency settles down toward the theoretical probability of 1/6
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Question: A fair die is rolled repeatedly, tracking the running relative frequency of rolling a six: after 20 rolls it is 0.25, after 40 rolls 0.2, after 60 rolls 0.183, after 80 rolls 0.175, and after 100 rolls 0.17. Describe the trend and what it demonstrates.
- Answer: The relative frequency is gradually settling down toward 1/6 ≈ 0.167 as the number of rolls increases - this is the Law of Large Numbers in action. It is not moving in a perfectly straight line, since chance still plays a role, but the overall trend is converging on the true probability.
Estimating a Population from Relative Frequency
If you know the total size of a population but not its make-up, you can estimate how many of each type it contains by multiplying the total by a sample's relative frequency.
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Question: A closed bag contains 60 counters, a mix of black and white in an unknown ratio. A counter is drawn at random, its colour recorded, then replaced; this is repeated 30 times, giving black 21 times. Estimate the number of black counters in the bag.
- Step 1: Relative frequency of black = 21 ÷ 30 = 0.7
- Step 2: Estimated number of black counters = 0.7 × 60
- Answer: 42 black counters (and an estimated 18 white counters)
Designing a Chance Experiment or Simulation
A good experiment uses enough trials for a reliable estimate, keeps conditions fair and consistent, and records results accurately.
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Question: A student wants to estimate the probability that a drawing pin lands "point up" when dropped. Describe a fair experiment design.
- Answer: Drop the same pin from the same height onto the same surface many times (e.g. 100+ times), recording point-up or point-down each time, then calculate relative frequency (point-up count ÷ total drops) as the estimated probability.
Real-World Applications
Comparing expected and observed results is used across many fields:
- Quality control: comparing expected vs observed defect rates in manufacturing.
- Sports analytics: comparing expected vs actual scoring rates.
- Insurance: comparing predicted vs actual claim frequencies.
- Gaming: checking whether games or devices are fair.
- Scientific experiments: comparing predicted vs observed results to test a hypothesis.
Exam Tips
Common Mistakes
MistakeAssuming any difference between expected and observed frequency means the experiment is unfair
Fixsome variation from expected is normal due to chance, especially with a small number of trials
MistakeAdding probability and trials instead of multiplying to find expected frequency
Fixexpected frequency = probability × number of trials
MistakeConfusing relative frequency with expected frequency
Fixrelative frequency comes FROM observed data (observed ÷ trials); expected frequency comes FROM the theoretical probability
MistakeTrusting a small number of trials as a reliable estimate of probability
Fixlarger numbers of trials generally give more reliable estimates (the Law of Large Numbers)
MistakeChanging experimental conditions between trials, e.g. how a coin is flipped
Fixkeep the method consistent across all trials for a fair, reliable experiment
MistakeAveraging two relative frequencies from different-sized experiments instead of pooling the counts
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