GCSE Probability scale and relative frequency Practice Questions

Count how many counters there are altogether

There are $3$ red, $5$ blue and $2$ green counters, so there are $10$ counters in the bag. Taking one at random makes all $10$ counters equally likely, and that is what lets us use theoretical probability.

Write the theoretical probability of red

Theoretical probability is the number of favourable outcomes divided by the total number of equally likely outcomes, so $P(\text{red}) = \frac{3}{10}$.

State the answer

The probability is $\frac{3}{10}$, written as a fraction in its simplest form.

Recall the two ends of the probability scale

A probability can never be smaller than $0$ (an event that cannot happen) or bigger than $1$ (an event that is certain). Every probability lies on the scale between them.

Say why the odd one out fails

$1.4$ is greater than $1$, and nothing can be more certain than certain.

State the answer

$1.4$ cannot be a probability.

Recall the two ends of the probability scale

A probability can never be smaller than $0$ (an event that cannot happen) or bigger than $1$ (an event that is certain). Every probability lies on the scale between them.

Write each number as a decimal so they can be compared

Fractions, decimals and percentages can all be probabilities, but they have to be written in the same form before they can be compared with $0$ and $1$.

Test each number against the scale

Four of the numbers lie between $0$ and $1$ inclusive, so each of them could be a probability. Only $\frac{7}{5}$ does not.

Place the possible probabilities on the scale

The four possible values all sit on the $0$ to $1$ scale, which is a quick visual check that they are legitimate probabilities.

Say why the odd one out fails

$\frac{7}{5}$ is greater than $1$, and nothing can be more certain than certain.

State the answer

$\frac{7}{5}$ cannot be a probability.

Say why the probabilities must be converted first

The probabilities are written in different forms. They cannot be put in order until they are all written in the same form, and decimals are the easiest.

Write the probability of event $A$ as a decimal

$0.15$ is already a decimal, so nothing needs doing.

Write the probability of event $B$ as a decimal

$\frac{2}{5}$ written as a decimal is $0.4$. Probabilities given as fractions, decimals and percentages cannot be compared until they are all in the same form.

Write the probability of event $C$ as a decimal

$70\%$ written as a decimal is $0.7$. Probabilities given as fractions, decimals and percentages cannot be compared until they are all in the same form.

Write the probability of event $D$ as a decimal

$\frac{1}{20}$ written as a decimal is $0.05$. Probabilities given as fractions, decimals and percentages cannot be compared until they are all in the same form.

Recall what the two ends of the scale mean

A probability of $0$ means the event cannot happen and a probability of $1$ means it is certain, so a bigger number always means a more likely event.

Mark the events on the probability scale

Marking each event on the $0$ to $1$ scale shows the order at a glance: the further to the right, the more likely the event.

Put the decimals in order, smallest first

Ordering the decimals from smallest to largest orders the events from least likely to most likely.

Read the events off in that order

The least likely event is $D$ and the most likely is $C$.

State the answer

In order from least likely to most likely: $D$, $A$, $B$, $C$

Recall that the probabilities of all the outcomes add up to 1

The spinner lands on exactly one of the four numbers every time, so the four probabilities must add up to exactly $1$.

Write the given probabilities over a common denominator

The lowest common denominator of the three given fractions is $8$, so rewrite each of them in that denominator before adding.

Add the given probabilities

The three given probabilities come to $\frac{5}{8}$.

Subtract from 1 to find the missing probability

Everything left over belongs to 4: $1 - \frac{5}{8} = \frac{3}{8}$.

Check the four probabilities add to 1

Adding the answer back on gives exactly $1$, which confirms it.

Check the missing probability is on the 0 to 1 scale

$\frac{3}{8}$ lies between $0$ (impossible) and $1$ (certain), so it is a possible probability.

Write the missing probability as a decimal

As a decimal the probability is $0.375$, which makes it easy to place on the probability scale and easy to multiply.

Write down the rule for the expected number of successes

Over many trials, an event with probability $p$ happens about $p$ of the time, so the expected number is the probability times the number of trials.

Substitute the probability and the number of spins

The probability is $\frac{3}{8}$ and the spinner is spun $320$ times.

Work out the expected number

$320 \div 8 \times 3 = 120$.

Check the expected number is sensible

The spinner cannot land on 4 more than $320$ times, and $120$ is well inside that range.

Say what kind of answer this is

This is an ESTIMATE. In a real set of $320$ spins the actual count would usually be near $120$ but not exactly $120$.

Link the estimate back to relative frequency

If the spinner really were spun $320$ times, the relative frequency of 4 would be close to $\frac{3}{8}$ - and closer still if it were spun more times. That is exactly how relative frequency estimates probability.

Note the mistake to avoid

Assuming each of the four numbers has probability $\frac{1}{4}$ would only be right for a FAIR spinner. This spinner is biased, so the probabilities have to be read from the question.

State the answer

An estimate for the number of times the spinner lands on 4 is $120$.