4.1.3 Expected Frequency
Expected frequency is a statistical term that represents the number of times an event or outcome is expected to occur in a particular situation, based on probability calculations or statistical models.
Expected frequency is often used in hypothesis testing, where researchers compare the observed frequencies of a certain event or outcome with the frequencies that would be expected if the null hypothesis were true.
The null hypothesis is a statement that assumes no significant difference or relationship between variables, and expected frequencies are used to test this assumption.
For example, if we conduct a survey and expect that \( 50 \) % of respondents will answer “yes” to a particular question, and we survey 100 people, we would expect \( 50 \) of them to answer “yes” based on probability calculations. If we observe that only \( 40 \) people answer “yes,” we can compare the observed frequency \( (40) \) with the expected frequency \( (50) \) to see if there is a statistically significant difference between them, which may indicate that the null hypothesis is incorrect.
To find the expected frequency for a particular event or outcome, you need to have some information about the probability of that event occurring and the total number of trials or observations.
Here is an example of how to find expected frequency.
If you are flipping a fair coin \( 100 \) times and you want to find the expected frequency of heads, you know that the probability of heads is \( 0.5 \).
Therefore, the expected frequency of heads would be:
\( Expected \ frequency \ of \ heads=probability \ of \ heads \times total \ number \ of \ trials \)
\( Expected \ frequency \ of \ heads=0.5 \times 100 \)
\( Expected \ frequency \ of \ heads=50 \)
So the expected frequency of heads is \( 50 \).
Test yourself
Question 1:
The diagram shows six discs.
Each disc has a color and a number.
One disc is picked at random.
Write down the probability that
the disc has the number \( 4 \),
[1]
the disc is red and has the number \( 3 \),
[1]
the disc is blue and has the number \( 4 \).
[1]
Question 2:
\( 20 \) students each record the mass, \( p \) grams, of their pencil case.
The table below shows the results.
A student is chosen at random.
Find the probability that this student has a pencil case with a mass greater than \( 150 \) g.
[1]
Question 3:
Sushila, has a bag which contains \( 10 \) red balls and \( 8 \) blue balls.
Sushila takes one ball at random from her bag.
Find the probability that she takes a red ball.
[1]
Question 4:
Tanya plants some seeds.
When a seed produces flowers, the probability that the flowers are red is \( 0.6 \) and the
probability that the flowers are yellow is \( 0.3 \).
Tanya has a seed that produces flowers.
Find the probability that the flowers are not red and not yellow.
[1]
Question 5:
There are \( 54 \) fish in a tank.
Some of the fish are white and the rest of the fish are red.
Jeevan takes at random a fish from the tank.
The probability that he takes a white fish is \( \frac{4}{9} \).
Work out the number of white fish originally in the tank.
[2]