Expected frequency
Have you noticed in probability that what you expected to happen can sometimes be different to what actually happens? In maths, we call this the expected frequency.
The expected frequency is how often we think an event will happen based on the theoretical probability or what has happened previously (relative frequency). For example, when we flip a coin, we would expect it to land on heads half the time. So if we flipped a coin $100$100 times, we would expect it to land on heads $50$50 times (because $\frac{50}{100}=\frac{1}{2}$50100=12).
To calculate the expected frequency of an event, we use the formula:
$\text{Expected Frequency of Event}=np$Expected Frequency of Event=np
where $n$n = the number times the experiment is repeated
$p$p = the probability of the event occurring
Worked example
A spinner is divided equally into $8$8 sections, but $3$3 of them are coloured green.
a) What is the probability of landing on green?
$\frac{\text{total favourable outcomes }}{\text{total possible outcomes }}=\frac{3}{8}$total favourable outcomes total possible outcomes =38
b) If the spinner is spun $145$145 times, how many times would you expect it to land on green?
We take the probability of the event P(green) and multiply it by the number of trials.
P(green) x $145$145 = $\frac{3}{8}\times145=54.375$38×145=54.375
At this point we need to round appropriately, so we could say that if the spinner is spun $145$145 times we could expect it to be green $54$54 times.
We can use the relative frequency of an event to calculate the expected frequency of an event that is not a repeatable procedure. For example, predicting the number of people of each blood type in a population given the percentage breakdowns.
Practice questions
Question 1
A car manufacturer found that $1$1 in every $4$4 cars they were producing had faulty brake systems. If they had already sold $5060$5060 cars, how many of those already sold would need to be recalled and repaired?
Question 2
Data collected in a certain town suggests that the probabilities of there being $0$0, $1$1, $2$2, $3$3, $4$4 or $5$5 or more car thefts in one day are as given in the table below.
What is the expected number of car thefts occurring on any particular day (to two decimal places)?
| Car Thefts | $0$0 | $1$1 | $2$2 | $3$3 | $4$4 | $5$5 or more |
| Probability | $0.09$0.09 | $0.24$0.24 | $0.20$0.20 | $0.16$0.16 | $0.18$0.18 | $0.13$0.13 |
Question 3
$260$260 standard six-sided dice are rolled.
What is the probability of getting an even number on a single roll of a die?
How many times would you expect an even number to come up on the $260$260 dice?
Question 4
$1000$1000 transistors were tested at a factory, and $12$12 were found to be faulty.
What is the experimental probability that a transistor at this factory will be faulty?
If another $5000$5000 transistors were tested, how many of these would you expect to be faulty?