Relative & Expected Frequency (Edexcel IGCSE Maths A (Modular))
Revision Note
Written by: Mark Curtis
Reviewed by: Dan Finlay
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Relative Frequency
What is relative frequency?
Relative frequency is an estimate of a probability using results from an experiment
For a certain number of trials of that experience, the probability of ‘success’ is:
If you flip an unfair coin 50 times and it lands on heads 20 times, an estimate for the probability of the coin landing on heads is (its relative frequency)
That is the best estimate we can make, given the data we have
We do not know the actual probability
The more trials that are carried out, the more accurate relative frequency becomes
It gets closer and closer to the actual probability
When will I be asked to use relative frequency?
Relative frequency is used when actual probabilities are unavailable (or not possible to calculate)
For example, if you do not know the actual probability of being left-handed, you can run an experiment to find an estimate (the relative frequency)
Sometimes actual probabilities are known, as they can be calculated in theory (called theoretical probabilities)
The theoretical probability of a fair coin landing on heads is 0.5
The theoretical probability of a fair standard six-sided dice landing on a six is
Relative frequency can be compared to a theoretical probability to test if a situation is fair or biased
If 100 flips of the coin give a relative frequency of 0.48 for landing on heads, the coin is likely to be fair
The theoretical probability is 0.5 and 0.48 is close to 0.5
If 100 flips of the coin give a relative frequency of 0.13 for landing on heads, the coin is likely to be biased (not fair)
What else do I need to know about relative frequency?
Relative frequency assumes that there is an equal chance of success on each trial
The trials are independent of each other
For example, if choosing something out of a bag (a ball, or marble etc), it would need to be replaced each time to use relative frequency
Any experiments used to calculate relative frequency should be random
If the experiment is not random, this could introduce bias
Examiner Tips and Tricks
Exam questions will not necessarily use the phrase relative frequency
If you have to choose the best estimate, choose the one with the most trials
Worked Example
There are an unknown number of different coloured buttons in a bag.
Johan selects a button at random, notes its colour and replaces the button in the bag.
Repeating this 30 times, Johan notes that on 18 occasions he selected a red button.
Use Johan’s results to estimate the probability that a button drawn at random from the bag is red.
Taking ‘red’ to be a success, Johan had 18 successes out of a total of 30 trials.
Expected Frequency
What is expected frequency?
Expected frequency refers to the number of times you would expect a particular outcome to occur
It is found by multiplying the probability by the number of trials
If you flip a fair coin 100 times, you would expect 0.5 × 100 = 50 heads
Sometimes you need to calculate the relative frequency first
If you flip a biased coin 40 times and get 10 heads, how many heads would you expect when flipping 100 times?
The relative frequency is = 0.25 from the first experiment
0.25 × 100 = 25, you would expect to get heads 25 times from 100 throws
Examiner Tips and Tricks
Exam questions will not necessarily use the phrase "expected frequency", but might ask how many you "would expect"
Worked Example
There are 6 blue, 4 red and 5 yellow counters in a bag.
One counter is drawn at random and its colour noted.
The counter is then returned to the bag.
(a) Find the probability that a counter drawn from the bag is yellow.
There are 5 yellow counters out of a total of 6 + 4 + 5 = 15 counters in the bag
P(Yellow)
(b) How many times would you expect a yellow counter to be drawn, if this experiment is repeated 300 times?
This is expected frequency so multiply the number of trials by the probability from part (a)
We would expect 100 yellow counters
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