Here are some more words to be familiar with in this topic:
An event is a set of outcomes for a random experiment. For example, flipping a coin and getting heads, tossing a die and getting an even number.
An experiment is a repeatable procedure with a set of possible outcomes. For example, tossing a die, flipping a coin.
Relative frequency
One way to predict the likelihood of an event happening is to investigate the occurrence of the event in the past. This is often what weather forecasters and economists do to predict what will happen in the future.
We can perform several trials of an experiment and observe the number of times the event occurs compared to all other events.
Relative frequency is a measure of the number of times that an event has occurred in a repeated experiment. We can write this as a fraction, ratio, decimal or percentage. Relative frequency is often referred to as experimental probability.
We calculate the relative frequency of an event by using the formula:
$\text{Relative Frequency }=\frac{\text{frequency of the event }}{\text{total number of trials }}$Relative Frequency =frequency of the event total number of trials
Worked example
$500$500 cables were tested at a factory, and $76$76 were found to be faulty.
a) What is the relative frequency of selecting a faulty cable at this factory?
$\frac{\text{frequency of the event }}{\text{total number of trials }}=\frac{76}{500}$frequency of the event total number of trials =76500
we can simplify this fraction to $\frac{19}{125}$19125, or convert it to a percentage which is $76\div500\times100=15.2%$76÷500×100=15.2%
b) If $1500$1500 more cables were tested, how many would you estimate to be faulty?
Now that we know that $15.2%$15.2% are faulty (from our experimental data), we could expect the same percentage to be faulty from any amount.
So, $15.2%$15.2% of $1500$1500
$\frac{15.2}{100}\times1500=228$15.2100×1500=228
We could expect $228$228 to be faulty from $1500$1500 cables.
Practice questions
Question 1
A coin was flipped $100$100 times with $51$51 tails recorded. What is the experimental probability of flipping tails with this coin? Give your answer in fully simplified form.
Question 2
The graph shows the four countries that university students applied to for exchange in the last month.
What is the relative frequency of the country with the fewest applications?
If the monthly applications are the same throughout the year, how many people in total will apply for UK over the next $12$12 months?
Question 3
A die is rolled $60$60 times and the results are recorded in the following table:
| Number | Frequency |
|---|---|
| $1$1 | $8$8 |
| $2$2 | $10$10 |
| $3$3 | $8$8 |
| $4$4 | $10$10 |
| $5$5 | $10$10 |
| $6$6 | $14$14 |
What is the relative frequency of rolling a $6$6 with this die? Express your answer in simplest form.
What is the relative frequency of rolling a $3$3 or higher with this die? Express your answer in simplest form.
What is the relative frequency of rolling a $3$3 or lower with this die? Express your answer in simplified form.
question 4
$16$16 dice were rolled and a $3$3 occurred $4$4 times.
What was the relative frequency of rolling a $3$3?
Using the results of the trial, predict how many times a $3$3 would occur if $64$64 dice are rolled?
Relative frequency and real world events
When we perform probability experiments, we are trying to model a real world event in order to determine the likelihood of certain events. Relative frequency will only provide an estimate for the probability of real world events, for example, weather forecasts and sports strategies. This is because it is often very difficult to simulate real world events accurately in order to predict or calculate the probability.
Class discussion
Identify some factors that can complicate the simulation of the following real-world events:
- Two equally matched football teams play $4$4 games each. Flip a coin $4$4 times to estimate the probability that the teams will win $2$2 games each.
- In a multiple choice test consisting of $20$20 questions, there are $4$4 options for each question with $1$1 of the options being the correct answer. Use a $4$4-sided die to estimate the probability of getting $5$5 of the questions correct.