Probability
Grade 4-5Relative Frequency
Relative frequency is what you actually observe happening in an experiment, an estimate of probability based on results rather than theory, and it becomes more trustworthy the more trials are carried out. This lesson covers calculating relative frequency, using it to estimate outcomes for a larger number of trials, and comparing it to theoretical probability to judge whether something is fair or biased.
What you need to know
- Relative frequency is the number of times something happened divided by the number of trials.
- It is an estimate of probability from an experiment, also called experimental probability.
- The more trials you do, the closer relative frequency gets to the true probability.
- Expected frequency is probability multiplied by the number of trials.
Written by Asad, co-founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
Calculating relative frequency
Relative frequency is calculated by dividing the number of times an event happened by the total number of trials.
A coin is flipped 40 times and lands on heads 18 times. The relative frequency of heads is 18 ÷ 40 = 0.45.
Using relative frequency to estimate an outcome
Relative frequency can be used to estimate how many times an event would happen in a larger number of trials, by multiplying the relative frequency by the new total.
A spinner lands on blue in 10 out of 50 spins, a relative frequency of 10/50. To estimate the number of blue results in 300 spins, multiply: (10/50) × 300 = 60.
Comparing relative frequency to theoretical probability
Theoretical probability is what should happen based on the maths, assuming everything is fair. Comparing an experiment's relative frequency to the theoretical probability shows whether the results are close to what was expected, or suggest bias.
A dice is rolled 60 times and lands on a 6 twenty times, a relative frequency of 20/60 ≈ 0.33. The theoretical probability of rolling a 6 is 1/6 ≈ 0.17. Since the relative frequency is much higher than the theoretical probability, the dice appears to be biased.
Why more trials give a more reliable estimate
A small number of trials can give a relative frequency far from the true probability just by chance. As the number of trials increases, relative frequency settles closer to the true probability, making the estimate more reliable.
A coin gives a relative frequency of 0.7 for heads after only 10 flips, but 0.51 after 500 flips. The estimate from 500 flips is more reliable, since a much larger number of trials brings the relative frequency closer to the true probability.
Worked examples
Three exam-style questions, fully solved
A coin is flipped 40 times and lands on heads 18 times. Calculate the relative frequency of heads.
Easy- 1.Identify the number of times the event happened: 18
- 2.Identify the total number of trials: 40
- 3.Divide the event count by the total: 18 ÷ 40
Answer: 0.45
A spinner is spun 50 times and lands on blue 10 times. Based on this, estimate how many times it would land on blue in 300 spins.
Medium- 1.Find the relative frequency of blue: 10 ÷ 50
- 2.Multiply the relative frequency by the new number of trials: (10/50) × 300
Answer: 60
A dice is rolled 60 times and lands on a 6 twenty times. The theoretical probability of rolling a 6 on a fair dice is 1/6. Comment on whether the dice appears to be fair.
Hard- 1.Calculate the relative frequency: 20 ÷ 60 ≈ 0.33
- 2.Convert the theoretical probability to a decimal: 1/6 ≈ 0.17
- 3.Compare the two values: 0.33 is much higher than 0.17
Answer: The dice appears to be biased, since the relative frequency is much higher than the theoretical probability
Practice
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Practice
Now try these yourself.
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A coin is flipped 200 times and lands on heads 90 times. Work out the relative frequency of heads, as a decimal.
A dice is rolled 60 times and shows a 6 on 12 of those rolls. Work out the relative frequency of rolling a 6, as a decimal.
After 500 trials of an experiment, the relative frequency of an event is 0.36. How many times did the event happen?
A spinner is spun 80 times. The relative frequency of landing on green is 0.25. How many times did it land on green?
A drawing pin is dropped 400 times and lands point-up 260 times. Use this to estimate the probability it lands point-up, as a decimal.
A biased dice is rolled 300 times and shows a 1 on 75 of them. Based on this, estimate how many 1s you would expect in 200 rolls.
Avoid these
The mistakes students make most often
Dividing the total number of trials by the frequency of the event, instead of the event's frequency by the total number of trials.
When estimating for a larger number of trials, adding the new total to the relative frequency instead of multiplying by it.
Comparing relative frequency and theoretical probability without converting both to the same form, such as decimals, making the comparison unclear.
Treating a single relative frequency, especially from a small number of trials, as if it were exactly the true probability, rather than just an estimate of it.
Stating that more trials are "more reliable" without explaining why, instead of linking it to relative frequency getting closer to the true probability as trials increase.
FAQ
Questions students and parents ask
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