Relative and expected frequencies
Section: Probability | Syllabus: Cambridge IGCSE Mathematics (0580)
Relative Frequency as an Estimate of Probability
- Relative frequency is used to estimate a probability from experimental data, especially when the theoretical probability is unknown or the outcomes are not equally likely: Relative frequency=(number of times the event occurs/total number of trials)
- Relative frequency is an estimate, not the true probability - repeating the same experiment again will usually give a slightly different relative frequency
- The more trials that are carried out, the more reliable relative frequency becomes as an estimate of the true probability
- Comparing a relative frequency to a known theoretical probability can suggest whether an object (such as a dice or coin) is biased
Worked Example: Using Relative Frequency to Test for Bias
- Question: A dice is rolled 200 times. The table shows the results.
Find the relative frequency of scoring a 6, and use it to comment on whether the dice appears to be fair.Score 1 2 3 4 5 6 Frequency 28 31 27 30 26 58 - Step 1: Relative frequency of scoring a 6 =(58/200)=0.29
- Step 2: For a fair dice, the theoretical probability of scoring a 6 is (1/6)0.167
- Step 3: 0.29 is considerably higher than 0.167
- Answer: The relative frequency of 0.29 is much greater than the fair-dice probability of (1/6), so the results suggest the dice is biased towards landing on 6
Common Mistakes
MistakeTreating a relative frequency calculated from a small number of trials as if it were the exact, true probability
Fixrelative frequency is only an estimate - it becomes more trustworthy as the number of trials increases, but even a large number of trials will not usually give the true probability exactly
MistakeDividing the frequency of one outcome by the frequency of another outcome instead of by the total number of trials
Fixthe denominator in relative frequency is always the total number of trials carried out, not the frequency of a different outcome
Expected Frequency
- Expected frequency predicts how many times an event should occur over a given number of trials: Expected frequency=probability of trials
- The probability used can be a theoretical probability, or an experimental estimate such as a relative frequency from previous trials
- Expected frequency is a prediction, not a guarantee - the actual number of occurrences in a real experiment will often differ from it
Worked Example: Calculating an Expected Frequency
- Question: Kabwe writes each of the numbers 2 to 15 on separate cards and puts them in a bag. He picks a card at random, notes the number, and replaces it.
- Step 1 (probability): Find the probability that the number is a factor of 24. The factors of 24 between 2 and 15 are 2,\,3,\,4,\,6,\,8,\,12 - that is 6 numbers out of 14 P(factor of 24)=(6/14)=(3/7)
- Step 2 (expected frequency): He repeats this 210 times. Find the expected number of times he picks a factor of 24 (3/7)210=90
- Answer: P(factor of 24)=(3/7); expected frequency =90
Common Mistakes
MistakeDividing the number of trials by the probability instead of multiplying
Fixexpected frequency is always probability multiplied by the number of trials, never the number of trials divided by the probability
MistakeExpecting the actual result of an experiment to match the expected frequency exactly
Fixexpected frequency is only a prediction based on probability - real experiments involve random variation, so the actual count is often close to, but not exactly, the expected frequency
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