Browse Basic probability

45 questions at your level

DifficultyAll Level 1 Level 2 Level 3 Level 4 Level 5

Probability of a single event

16 questions

LessonNot started

A fair six-sided dice is rolled. Write down the probability of rolling a

4

●●●●●Level 11 markStart

The probability that it rains tomorrow is

0.3

. Write down the probability that it does not rain tomorrow.

●●●●●Level 11 markStart

A bag contains 5 red counters, 8 blue counters and 12 green counters.

A counter is taken at random from the bag.

(a) Write down the probability that the counter is blue.

(b) Work out the probability that the counter is not green.

●●●●●Level 23 marksStart

A biased spinner can land on red, blue, green or yellow.

The table shows the probability that the spinner lands on each of three of these colours.

Colour	Red	Blue	Green	Yellow
Probability	0.2	0.35	0.15

(a) Work out the probability that the spinner lands on yellow.

(b) Work out the probability that the spinner does not land on blue.

●●●●●Level 23 marksStart

A bag contains

3

red counters,

5

blue counters and

4

green counters.

A counter is taken at random from the bag.

(a) Work out the probability that the counter is not green. (1 mark)

The counter is put back in the bag. This is repeated

60

times.

(b) Work out an estimate for the number of times a green counter is taken. (2 marks)

●●●●●Level 23 marksStart

80

students each study exactly one language: French, Spanish or German.

45

of the students are girls.

20

students study Spanish, and

8

of these are girls.

25

girls study French.

12

boys study German.
(a) Work out how many boys study French. (2 marks)
(b) One of the students who studies German is chosen at random.

Work out the probability that this student is a girl. (1 mark)

●●●●●Level 23 marksStart

A fair six-sided dice has faces numbered 1, 2, 3, 4, 5 and 6.

(a) Write down the probability that the dice lands on a number greater than 4.

The dice is rolled 180 times.

(b) Work out an estimate for the number of times the dice lands on a number greater than 4.

●●●●●Level 33 marksStart

A bag contains only red beads, blue beads and green beads.

A bead is taken at random from the bag.

The probability that the bead is red is

3x

.

The probability that the bead is blue is

2x

.

The probability that the bead is green is

x

.

(a) Work out the value of

x

.

(b) Work out the probability that the bead is blue.

●●●●●Level 33 marksStart

There are

30

students in a class.

18

of them are girls.

12

of the students wear glasses.

7

of the girls wear glasses.

A student is chosen at random. Work out the probability that the student is a boy who does not wear glasses. (3 marks)

●●●●●Level 33 marksStart

There are only gold tokens, silver tokens, bronze tokens and copper tokens in a box.

Amir is going to take at random a token from the box.

The table shows the probability that he will take a gold token and the probability that he will take a silver token.

Token	gold	silver	bronze	copper
Probability	$0.25$	$0.15$

There are three times as many bronze tokens as copper tokens in the box.
(a) Work out the probability that Amir will take a bronze token.

(3 marks)

There are

36

silver tokens in the box.
(b) Work out the total number of tokens in the box.

(2 marks)

●●●●●Level 35 marksStart

There are only green marbles, blue marbles and white marbles in a tin.

There are

18

white marbles in the tin.

Priya is going to take at random a marble from the tin.

The probability that the marble will be white is

0.15

number of green marbles : number of blue marbles

= 7 : 10

Work out the number of green marbles in the tin.

(4 marks)

●●●●●Level 34 marksStart

A biased four-sided spinner can land on 1, 2, 3 or 4.

The table shows information about the probability that the spinner lands on each number. The spinner is equally likely to land on 3 as it is to land on 4.

Number	1	2	3	4
Probability	0.15	0.35	$y$	$y$

(a) Work out the value of

y

.

The spinner is spun 200 times.

(b) Work out an estimate for the number of times the spinner lands on an even number.

●●●●●Level 46 marksStart

Priya plays one game of chess each week. Each game she either wins, draws or loses.

The probability that Priya wins a game is

\frac{2}{5}

.

The probability that Priya draws a game is

15\%

.

(a) Work out the probability that Priya loses a game. Give your answer as a decimal.

(b) Priya plays 60 games. Work out an estimate for the number of these games that Priya loses.

●●●●●Level 45 marksStart

A bag contains only red counters and yellow counters.

There are 12 red counters in the bag.

A counter is taken at random from the bag. The probability that it is red is

0.3

.

(a) Work out the total number of counters in the bag.

(b) Work out the number of yellow counters in the bag.

A counter is taken at random, its colour is recorded, and the counter is put back in the bag. This is done 250 times.

(c) Work out an estimate for the number of times a yellow counter is taken.

●●●●●Level 45 marksStart

There are only

3

purple pens and

5

orange pens in a box.

Jay takes at random a pen from the box.

He records its colour and puts the pen back into the box.

Jay does this two more times.

Show that the probability that Jay takes at least two purple pens is

\frac{81}{256}

(4 marks)

●●●●●Level 44 marksStart

A bag contains only blue beads, green beads and white beads.

A bead is taken at random from the bag.

The probability that the bead is blue is

\frac{1}{4}

.

The probability that the bead is green is three times the probability that the bead is white.

(a) Work out the probability that the bead is white.

There are 96 beads in the bag.

(b) Work out the number of green beads in the bag.

Alex takes a bead at random, records its colour and puts it back. He does this 96 times.

Alex says, "The number of times I record a green bead will be exactly the same as the number of green beads in the bag."

(c) Explain why Alex is wrong.

●●●●●Level 57 marksStart

Sample space & systematic listing

16 questions

LessonNot started

A coin is flipped and a fair four-sided dice (numbered

1

4

) is rolled. List all the possible outcomes.

●●●●●Level 11 markStart

Two fair coins are flipped. Write down the probability of getting two heads.

●●●●●Level 12 marksStart

A menu has

3

starters and

4

main courses. How many different combinations of one starter and one main are there?

●●●●●Level 11 markStart

A fair coin is flipped and a fair four-sided spinner numbered

1

2

3

4

is spun.

(a) List all the possible outcomes.

(b) Find the probability of getting a head and an even number.

●●●●●Level 24 marksStart

Spinner A has four equal sections numbered

1, 2, 3, 4

. Spinner B has three equal sections numbered

1, 2, 3

.

Both spinners are spun and the scores are added.

Work out the probability that the total is

5

. [3]

●●●●●Level 23 marksStart

Two fair six-sided dice are rolled and the two scores are added to give a total.

(a) Find the probability that the total is

5

.

(b) Find the probability that the total is at least

10

.

(c) Jay says: "The total can be any whole number from

2

12

, so there are

11

possible outcomes and the probability of a total of

5

\frac{1}{11}

."

Explain why Jay is wrong.

●●●●●Level 35 marksStart

Two fair six-sided dice are rolled. The score is the positive difference between the two numbers rolled. For example, a

6

and a

2

give a score of

4

, and a

3

and a

3

give a score of

0

.

(a) Find the probability that the score is

1

.

(b) Find the probability that the score is

3

or more.

(c) Write down the most likely score, and explain how the sample space shows this.

●●●●●Level 36 marksStart

A security code is made from one letter chosen from A, B, C, D, followed by two digits, each chosen from

0

9

.

For example, C07 is a possible code.

(a) Work out how many different codes are possible.

(b) Work out how many of these codes have two digits that are different from each other.

(c) One of the codes from part (a) is chosen at random. Find the probability that it ends in

00

●●●●●Level 46 marksStart

A club has six members: Ana, Ben, Cal, Dee, Eve and Femi.

(a) Two of the six members are chosen to go to a conference together. Work out how many different pairs could be chosen.

(b) Instead, one of the six is chosen as chairperson and a different one is chosen as secretary. Work out how many different ways this can be done.

(c) Explain how the answer to part (b) is related to the answer to part (a).

●●●●●Level 44 marksStart

A fair six-sided die numbered

1

6

is rolled, and a fair four-sided spinner numbered

1

2

3

and

5

is spun.

The number on the die is multiplied by the number on the spinner to give the score.

(a) Work out how many equally likely outcomes there are.

(b) Find the probability that the score is an odd number.

(c) Find the probability that the score is greater than

20

●●●●●Level 45 marksStart

Each sandwich at a sandwich bar is made with one type of bread, one filling and one sauce.

There are

4

types of bread.

There are

9

fillings.

There are

x

sauces.

There are

252

different ways to choose one type of bread, one filling and one sauce.

Work out the value of

x

.

(2 marks)

●●●●●Level 42 marksStart

There are

11

runners in a junior race.

One runner wins a gold medal.

A different runner wins a silver medal.

There are

14

runners in a senior race.

One runner wins a gold medal.

A different runner wins a silver medal.

Work out how many different ways these four medals can be won.

(3 marks)

●●●●●Level 43 marksStart

A four-digit code is made using these rules:

the first digit must be odd,
the last digit must be $0$ or $5$ ,
the second and third digits can be any digit from $0$ to $9$ .

Digits may be repeated.

(a) Work out how many different codes are possible.

(b) A code is chosen at random from all the possible codes. Find the probability that all four of its digits are the same.

●●●●●Level 55 marksStart

In a football league every team plays every other team exactly once. Altogether

45

matches are played.

(a) Work out how many teams are in the league.

(b) One of the

45

matches is chosen at random. Rovers is one of the teams in the league.

Find the probability that the chosen match is one that Rovers plays in.

●●●●●Level 55 marksStart

There are only

5

white counters,

3

purple counters and

2

black counters in a bag.

Omar takes at random three counters from the bag.

Work out the probability that the bag is then left with more black counters than purple counters.

You must show all your working.

(5 marks)

●●●●●Level 55 marksStart

There are

12

discs numbered

1

12

in a tin.

Hana will take at random two discs from the tin.

Hana says,

"The probability that the sum of the two numbers is odd is twice the probability that their product is odd."

Is Hana correct?

You must show how you get your answer.

(5 marks)

●●●●●Level 55 marksStart

Relative frequency & expected outcomes

14 questions

LessonNot started

A spinner is spun

50

times. It lands on red

20

times. Work out the relative frequency of red.

●●●●●Level 11 markStart

The probability that a biased coin lands heads is

0.6

. The coin is flipped

200

times. Estimate how many times it lands heads.

●●●●●Level 11 markStart

Sam drops a drawing pin

200

times. It lands point-up

130

times.

(a) Use Sam's results to estimate the probability that the pin lands point-up.

(b) Estimate the probability that the pin lands point-down.

●●●●●Level 23 marksStart

A bag contains

3

red counters,

5

blue counters and

4

green counters.

A counter is taken at random from the bag.

(a) Work out the probability that the counter is not green. (1 mark)

The counter is put back in the bag. This is repeated

60

times.

(b) Work out an estimate for the number of times a green counter is taken. (2 marks)

●●●●●Level 23 marksStart

A dice is rolled

200

times. It lands on

6

44

of the rolls.

(a) Work out the relative frequency of rolling a

6

. (1 mark)

(b) The dice is rolled

500

times. Estimate the number of times it lands on

6

. (1 mark)

(c) Do you think the dice is fair? Give a reason for your answer. (1 mark)

●●●●●Level 23 marksStart

A spinner has four equal sections coloured red, blue, green and yellow.

The spinner is spun

80

times. Here are the results.

Colour	Red	Blue	Green	Yellow
Frequency	28	20	12	20

(a) Work out the relative frequency of red. (1 mark)
(b) Is the spinner fair? Give a reason for your answer. (1 mark)
(c) The spinner is spun

400

times. Estimate the number of times it will land on red. (1 mark)

●●●●●Level 23 marksStart

A spinner has four colours on it. Priya spins the spinner

60

times and records the colour it lands on.

\begin{array}{l|cccc}\text{Colour} & \text{Red} & \text{Blue} & \text{Green} & \text{Yellow} \\ \hline \text{Frequency} & 24 & 15 & 9 & 12\end{array}

(a) Use Priya's results to estimate the probability that the spinner lands on green.

(b) Priya spins the spinner a further

400

times. Estimate the number of times it lands on red.

●●●●●Level 34 marksStart

Jamal rolls a dice

300

times. He rolls a six

72

times.

(a) Work out the relative frequency of rolling a six.

(b) Work out how many sixes Jamal would expect in

300

rolls if the dice were fair.

(c) Do Jamal's results suggest that the dice is biased? Give a reason for your answer.

●●●●●Level 34 marksStart

In a computer game, Leo attempts a level

80

times and completes it

28

times.

(a) Use these results to estimate the probability that Leo completes the level on any one attempt.

(b) Leo attempts the level

250

more times. Estimate the number of times he completes it.

(c) Explain why your answer to part (b) cannot be the actual number of times Leo completes the level.

●●●●●Level 35 marksStart

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 43 marksStart

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 44 marksStart

A spinner has five equal sections numbered

1

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

5

, and is known to be fair. Estimate the number of

1

s in

3000

spins of this spinner.

●●●●●Level 46 marksStart

A biased spinner has five sections numbered

1

2

3

4

and

5

Leah spins the spinner

80

times.

The table gives the number of times the spinner lands on each number.

Number	$1$	$2$	$3$	$4$	$5$
Frequency	$14$	$20$	$11$	$19$	$16$

Leah is going to spin the spinner two more times.
(a) Work out an estimate for the probability that the spinner will land on

4

both times.

(3 marks)

Marcus will spin the same spinner

n

times and use his results to get a better estimate of this probability.
(b) What can you say about the value of

n

?

(1 mark)

●●●●●Level 44 marksStart

A company tests a sample of

500

of its light bulbs.

12

of them fail within a year.

(a) Use this sample to estimate the probability that a bulb fails within a year.

(b) The company sells

25\,000

bulbs. Every bulb that fails within a year is replaced, at a cost of £

3.50

to the company. Estimate the total replacement cost.

(c) The company then tests a second sample of

1500

bulbs, of which

27

fail within a year. Use both samples to give a better estimate of the probability that a bulb fails within a year.

(d) Explain why your answer to part (c) is a better estimate than your answer to part (a).

●●●●●Level 58 marksStart