Experimental Probability and Relative Frequency
Experimental probability, also called relative frequency, estimates how likely an event is using real data from trials rather than by assuming outcomes are equally likely, calculated as the number of times the event happened divided by the total number of trials. It is used when an object might be biased, or when a theoretical probability cannot be worked out directly, and generally becomes more reliable, closer to the true probability, the more trials are carried out.
Before you start
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Method
- Identify the number of times the event of interest occurred and the total number of trials carried out.
- Calculate the relative frequency as (number of successful trials) divided by (total number of trials), giving a fraction, decimal or percentage.
- If a theoretical probability is available (for example, 1/6 for a fair dice), compare it with the relative frequency to judge whether the object might be biased.
- Remember that a small number of trials can give a relative frequency quite different from the true probability, purely by chance, so more trials generally give a more reliable estimate.
- To predict the number of times an event will happen in a future number of trials, multiply the relative frequency (or a given probability) by that number of trials.
- When results are given for several different numbers of trials, use the relative frequency from the LARGEST number of trials as the best estimate of the true probability.
Worked example
A drawing pin is dropped 200 times. It lands point-up 74 times. (a) Calculate the relative frequency of the pin landing point-up. (b) Estimate how many times the pin would land point-up in 500 drops.
- Identify the values: the pin landed point-up 74 times out of 200 total drops.
- Calculate the relative frequency: 74/200 = 0.37.
- This is the best available estimate of the true probability of the pin landing point-up.
- To predict the number of point-up landings in 500 drops, multiply the relative frequency by 500: 0.37 x 500.
- 0.37 x 500 = 185, so the pin would be expected to land point-up approximately 185 times in 500 drops.
Practice questions
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Q1A biased dice is rolled 50 times and lands on a six 20 times. Calculate the relative frequency of rolling a six.Show answer
Answer: 0.4 (20/50)
Q2A netball player takes 40 shots in training and scores 28 of them. Calculate the relative frequency of her scoring a shot, as a percentage.Show answer
Answer: 70% (28/40)
Q3A spinner is spun 80 times and lands on red 15 times. Estimate the probability that the spinner lands on red on the next spin.Show answer
Answer: 0.1875 (15/80), which can also be written as 3/16
Q4A coin is flipped 10 times and lands on heads 8 times. Explain why this result does not necessarily mean the coin is biased.Show answer
Answer: 10 trials is a small sample, so the result could easily be due to chance rather than bias; a much larger number of trials would be needed to reliably estimate whether the coin is biased.
Q5A factory tests 300 light bulbs and finds that 6 are faulty. Estimate how many faulty bulbs there would be in a batch of 5000.Show answer
Answer: 100 (relative frequency = 6/300 = 0.02, and 0.02 x 5000 = 100)
Q6A four-sided spinner is tested and the results recorded after different numbers of spins: after 20 spins it lands on blue 7 times; after 500 spins it lands on blue 142 times. Which relative frequency gives the better estimate of the true probability of landing on blue, and why?Show answer
Answer: The 500-spin result (142/500 = 0.284), because a larger number of trials generally gives a more reliable estimate of the true probability.
Q7A town's weather records show that, of the last 40 Christmas Days, it snowed on 6 of them. Use this data to estimate the probability that it will snow in the town this Christmas Day.Show answer
Answer: 0.15 (6/40 = 3/20)
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
A four-sided spinner, numbered 1 to 4, is suspected of being biased. It is spun 250 times, with these results: landed on 1 a total of 40 times, landed on 2 a total of 45 times, landed on 3 a total of 115 times, landed on 4 a total of 50 times. (a) Calculate the relative frequency of landing on 3. (b) Give one reason why the spinner is likely to be biased.
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A dice is rolled 60 times, with these results: score 1, 8 times; score 2, 9 times; score 3, 11 times; score 4, 10 times; score 5, 9 times; score 6, 13 times. (a) Calculate the relative frequency of rolling a 6. (b) The dice is rolled a further 240 times, making 300 rolls in total. Using the relative frequency from part (a), estimate the total number of times a 6 would be rolled in all 300 rolls.
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Two different spinners, A and B, are each spun 400 times to see how often they land on red. Spinner A lands on red 92 times. Spinner B lands on red 122 times. (a) Calculate the relative frequency of landing on red for each spinner. (b) A fair spinner divided into 4 equal sections, one of which is red, would be expected to land on red with probability 0.25. Use your answers to decide which spinner, if either, appears to behave like a fair 1-in-4 spinner, and explain your reasoning.
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Free printable worksheet
Want more practice on paper? Download the experimental probability and relative frequency worksheet pack - 6 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 24 of KS3 Maths Workbook 2, the whole course as one free printable PDF.
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