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Grade 5 · Foundation and Higher · free GCSE maths worksheet

Tree diagrams for independent events

Multiplying along branches and adding between.

18 original questions, 63 marks. Print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.

Worksheet PDFWorked solutions PDFAnswers PDF

The questions

1

The probability that it rains in a town on Saturday is 0.3.

The probability that it rains in the town on Sunday is 0.4.

The two events are independent.

Work out the probability that it rains on both days.

2 marks
Show answer
0.12
2

Zak plays two games of darts. He draws a probability tree diagram.

On the first pair of branches he writes P(win) = 0.6 and P(lose) = 0.5.

He then works out P(win both games) = 0.6 + 0.6 = 1.2.

Write down two mistakes Zak has made.

The games are independent, with the same probability of winning each game. There are no draws.

2 marks
Show answer
The first pair of branch probabilities do not add up to 1 (0.6 + 0.5 = 1.1); he should have multiplied along the branches, not added (and a probability cannot be more than 1).
3

Femi plays two games of table tennis.

The probability that he wins each game is 23.

The results of the games are independent.

Work out the probability that Femi loses at least one of the games.

3 marks
Show answer
59
4

To win a chess tournament, Jamal must win his semi-final and then win the final.

The probability that he wins his semi-final is 0.7.

If he reaches the final, the probability that he wins it is 0.55.

Work out the probability that Jamal wins the tournament.

2 marks
Show answer
0.385
5

A bag contains 3 red counters and 7 blue counters.

Kira takes a counter at random, notes its colour and puts it back.

She then takes a second counter at random.

  1. Complete the probability tree diagram for the two picks.
  2. Work out the probability that both counters are blue.
First counterSecond counterRedRedBlueBlueRedBlue
3 marks
Show answer
a) P(red) = 0.3 and P(blue) = 0.7 on every pair of branches b) 0.49
6

The probability that Ravi’s bus is on time on any day is 0.8.

Whether the bus is on time on one day does not affect any other day.

Ravi catches the bus on Monday and on Tuesday.

  1. Work out the probability that the bus is on time on both days.
  2. Work out the probability that the bus is late on exactly one of the two days.
3 marks
Show answer
a) 0.64 b) 0.32
7

The probability that Jon's bus is on time on any day is 0.9.

Whether the bus is on time on one day is independent of any other day.

Jon catches the bus on Monday and on Tuesday.

  1. Complete the probability tree diagram.
  2. Work out the probability that the bus is on time on both days.
MondayTuesday0.9On timeOn timeLateLateOn timeLate
4 marks
Show answer
a) P(late) = 0.1 on each pair of branches, P(on time) = 0.9 on the second day branches b) 0.81
8

A bag contains 4 red counters and 6 blue counters.

Eva takes a counter at random, records its colour and puts it back. She then takes a second counter at random.

  1. Work out the probability that both counters are red.
  2. Work out the probability that exactly one of the counters is red.
3 marks
Show answer
a) 0.16 b) 0.48
9

Amy and Beth each take one penalty kick.

The probability that Amy scores is 0.7.

The probability that Beth scores is 0.55.

The two events are independent.

Work out the probability that neither of them scores.

4 marks
Show answer
0.135
10

Ella and Fraser each take their driving test on the same day. Their results are independent.

The probability that Ella passes is 0.75.

The probability that Fraser passes is 0.6.

  1. Work out the probability that both pass.
  2. Work out the probability that exactly one of them passes.
  3. Work out the probability that at least one of them passes.
4 marks
Show answer
a) 0.45 b) 0.45 c) 0.9
11

A bag contains 2 green sweets and 3 yellow sweets.

Cal takes a sweet at random, notes its colour and puts it back in the bag.

He then takes a second sweet at random.

Work out the probability that the two sweets are the same colour.

4 marks
Show answer
1325
12

Zainab takes three free throws in a basketball game. The probability that she scores with any one throw is 0.7, and the throws are independent.

  1. Work out the probability that she scores with all three throws.
  2. Work out the probability that she misses at least once.
  3. Work out the probability that she scores with at least two of the three throws.
4 marks
Show answer
a) 0.343 b) 0.657 c) 0.784
13

A warehouse has two independent fire alarms.

The probability that each alarm works correctly in a test is 0.95.

Both alarms are tested.

Work out the probability that at least one of the alarms works correctly.

4 marks
Show answer
0.9975
14

A gardener sows three seeds. Each seed has probability 0.9 of germinating, independently of the others.

Work out the probability that exactly two of the three seeds germinate.

3 marks
Show answer
0.243
15

Cara and Dan each take a driving test.

The probability that Cara passes is 0.6.

The probability that Dan passes is 0.75.

The two events are independent.

Work out the probability that exactly one of them passes.

4 marks
Show answer
0.45
16

A biased coin is flipped twice. The flips are independent.

The probability of getting tails on both flips is 0.36.

  1. Work out the probability that the coin lands on heads on one flip.
  2. Work out the probability of getting exactly one tail in the two flips.
4 marks
Show answer
a) 0.4 b) 0.48
17

Ella plays a tennis match on Saturday and a badminton match on Sunday.

The probability that she wins the tennis match is 0.65.

The probability that she wins the badminton match is 0.4.

The results are independent.

  1. Complete the probability tree diagram.
  2. Work out the probability that Ella wins exactly one of the two matches.
TennisBadminton0.65WinWinLoseLoseWinLose
5 marks
Show answer
a) P(lose tennis) = 0.35; badminton branches 0.4 win and 0.6 lose from each tennis outcome b) 0.53
18

A box contains 5 red pens and 4 blue pens.

Milo takes a pen at random and does not replace it.

He then takes a second pen at random.

  1. Complete the probability tree diagram.
  2. Work out the probability that Milo takes at least one red pen.
First penSecond penRedRedBlueBlueRedBlue
5 marks
Show answer
a) first pick 5/9 and 4/9; second pick 4/8, 4/8 after red and 5/8, 3/8 after blue b) 5/6

Every question here is original and written by Robinson Tuition. The grade follows the Edexcel grade descriptors and suits AQA and OCR too; this is not an exam-board paper.

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