GCSE · Basic probability

Relative frequency & expected outcomesExam-style questions

Estimating a probability from data — relative frequency is frequency divided by the number of trials — why more trials sharpen the estimate, how to test a dice or spinner for bias, and why the expected number is probability times trials.

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13 questions · GCSE (Higher) level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 1 Level 2 Level 3 Level 4 Level 5

1
There are some red, blue, green and yellow cubes in a box.

Kwame takes a cube at random from the box, writes down its colour and puts the cube back in the box.

He does this $50$ times.

The table shows his results.
Colour red blue green yellow
Frequency $13$ $9$ $16$ $12$
Kwame is now going to take a cube at random, put it back, and then take another cube at random.
(a) Work out an estimate for the probability that both cubes will be green.

(3 marks)

Leila repeats Kwame's experiment with the same box, but she takes a cube $n$ times.

Her results give a more reliable estimate of the probability of taking a green cube than Kwame's results.
(b) What can you say about the value of $n$ ?

(1 mark)
Level 4 · 4 marksTry it
2
Ann and Ben each test the same spinner.

Ann spins it $50$ times and it lands on green $18$ times.

Ben spins it $150$ times and it lands on green $42$ times.

(a) Use all of their results to estimate the probability that the spinner lands on green.

(b) Chloe says the best estimate is the mean of Ann's and Ben's relative frequencies, which is $0.32$ . Explain why Chloe's method does not give the best estimate.
Level 4 · 3 marksTry it
3
A spinner can land on red, blue or white. Nadia spins the spinner a number of times. The relative frequency of red is $0.28$ , and the spinner landed on red $63$ times.

(a) Work out how many times Nadia spun the spinner.

(b) The spinner landed on blue $90$ times. Use Nadia's results to estimate the probability that the spinner lands on white.
Level 4 · 4 marksTry it
4
A spinner has five equal sections numbered $1$ to $5$ . Priya spins it $500$ times, working out the relative frequency of a $1$ as she goes.

$\begin{array}{l|cccccc}\text{Number of spins} & 10 & 20 & 50 & 100 & 200 & 500 \\ \hline \text{Relative frequency of a }1 & 0.3 & 0.25 & 0.22 & 0.21 & 0.205 & 0.202\end{array}$

(a) Which of Priya's figures is the best estimate of the probability of spinning a $1$ ? Give a reason for your answer.

(b) Do Priya's results suggest the spinner is fair? Explain your answer.

(c) A different spinner also has five equal sections numbered $1$ to $5$ , and is known to be fair. Estimate the number of $1$ s in $3000$ spins of this spinner.
Level 4 · 6 marksTry it