Estimating probability
In an experiment or survey, relative frequency of an event is the number of times the event occurs divided by the total number of trials.
For example, if you observed \(100\) passing cars and found that \(23\) of them were red, the relative frequency would be \(\frac{23}{100}\).
Accuracy
The Probability guide describes how you can get a more accurate result in surveys of events if you carry out a large number of trials or survey a large number of people.
Example
This bag contains \({3}\) red sweets and \({7}\) blue sweets.
Tom took a sweet from the bag, noted its colour and then replaced it.
He did this \(10\) times and found that he obtained a red sweet on \(4\) occasions, so the relative frequency of the event that a red sweet was chosen is \(\frac{4}{10}\).
He then carried out the experiment another \({10}\) times and combined his results with the first trial.
He found that he had obtained a red sweet on \(5\) out of \(20\) occasions, so the relative frequency of the event that a red sweet was chosen was \(\frac{5}{20}\).
Tom continued in this way, recording his combined results after every \({10}\) trials and plotting them on the graph below:
We can see from the graph that relative frequency gets better (ie closer to the true probability) as the number of trials increases.
Calculating relative frequency
Try out these example questions about relative frequency.
Question
Q1. \(100\) people were asked if they were left-handed. Four people answered 'yes'. What is the relative frequency of left-handed people?
Q2. A white counter was taken from a bag of different coloured counters, and then replaced. The relative frequency of getting a white counter was found to be \(0.3\).
If the bag contained \(20\) counters, estimate the number of white counters.
Answers
A1. The relative frequency is:
\(\frac{4}{100}\) = \(\frac{1}{25}\) or \(0.04\)
A2. The relative frequency of white counters is \(0.3\), and there are \(20\) counters in the bag.
So, as an estimate, \(0.3 \times 20 = 6\) are white counters.
Key point
Relative frequency can only be used as an estimate for a true probability.
Test section
Question 1
What is frequency?
a) Frequency = number of times the event happens 梅 total number of trials
b) Frequency = number of times the event happens
c) Frequency = total number of trials
Answer
You only need to count to find frequency, so the answer is Frequency = number of times the event happens.
Question 2
To calculate relative frequency we have to use the formula:
a) Relative frequency = total number of trials
b) Relative frequency = number of times the event happens
c) Relative frequency = number of times the event happens 梅 total number of trials
Answer
You need to divide by the total number of trials to get relative frequency - so the correct answer is Relative frequency = number of times the event happens 梅 total number of trials.
Question 3
The results of rolling a conventional \({6}\)-sided die are:
\( {6} \), \( {3} \), \( {4} \), \( {3} \), \( {5} \), \( {5} \), \( {2} \), \( {1} \), \( {2} \), \( {5} \), \( {6} \), \( {4} \), \( {3} \), \( {1} \), \( {2} \), \( {4} \), \( {5} \), \( {5} \), \( {3} \), \( {6} \), \( {3} \), \( {4} \), \( {1} \), \( {4} \).
What is the frequency of the number \(4\)?
Answer
You need to count how many times the number \({4}\) appears in the list to get its frequency - which is \({5}\).
Question 4
The results of rolling a conventional \({6}\)-sided die are:
\({6}\), \({3}\), \({4}\), \({3}\), \({5}\), \({5}\), \({2}\), \({1}\), \({2}\), \({5}\), \({6}\), \({4}\), \({3}\), \({1}\), \({2}\), \({4}\), \({5}\), \({5}\), \({3}\), \({6}\), \({3}\), \({4}\), \({1}\), \({4}\).
What's the relative frequency of the number \({4}\)?
Answer
\({5}\) is the frequency.
You need to divide this by the number of trials to get the relative frequency.
The answer is \(\frac{5}{24}\).
Question 5
What's the smallest possible value for relative frequency?
Answer
The smallest possible frequency is \({0}\) and if you divide it by the number of trials, the answer will still be \({0}\).
Question 6
What's the biggest possible value for relative frequency?
Answer
The greatest possible frequency is equal to the number of trials.
So if there are \({20}\) trials, the greatest possible frequency is \({20}\), and the greatest possible relative frequency is \({20}\div{20}={1}\)
Question 7
When two coins are tossed a large number of times, what is the approximate relative frequency of getting TT ('tails' and 'tails')?
Answer
There are four possibilities that are equally likely, namely HH, TH, HT and TT.
So we can expect TT about a quarter of the time or \({0.25}\) times.
Question 8
Here are the outcomes of tossing two coins \({10}\) times (where T means 'tails' and H means 'heads'):
TH, HH, HT, HH, HT, TT, HT, TH, TH, HH
What's the relative frequency of HH?
Answer
\({3}\) is the frequency of HH (it occurs \({3}\) times) and the number of trials is \({10}\).
So the relative frequency \(={3}\div{10}={0.3}\)
Question 9
A bag contains \({100}\) counters of different colours.
Lois pulls a counter from the bag, notes its colour, and then puts it back in the bag.
Lois does this a large number of times.
The relative frequency of the colour red is \({0.4}\).
What is a good estimate for the number of red counters in the bag?
Answer
Over a large number of trials, the relative frequency gets closer to the probability of selecting a red counter.
So, the number of red counters = \({relative~frequency}\times{number~of~trials}\) \(={0.4}\times{100}={40}\).
Question 10
Jacob records the weather over a long period of time and observes that it rains on \({45}\%\) of the days.
Assuming that this is constant throughout the year, about how many of the next \({40}\) days will see rain?
Answer
To get the answer you need to multiply the relative frequency by the number of days, that is, multiply \({0.45}\times{40}={18}\).
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