Probability (Higher) Worksheet with Answers

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GCSE · Statistics and Probability

4.19 Probability (Higher)

EDEXCEL 1MA1 · Calculator allowed · about 105 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

1

This question is about basic probability.

(a)An event is described as 'unlikely'. Which of these values could be the probability of this event?(1)

A) 0.02
B) 0.15
C) 0.6
D) 0.95

(b)A fair six-sided dice, with faces numbered 1 to 6, is rolled once. Write down the probability that the score is a multiple of 3.(1)

(c)Write down the probability that the score is not a multiple of 3.(1)

(Total for Question 1 is 3 marks)

2

Fatima has a biased coin. She flips it 40 times and records 28 heads.

(a)Calculate the relative frequency of heads.(1)

(b)Fatima is going to flip the coin 500 times. Estimate the number of times it will land on heads.(2)

(Total for Question 2 is 3 marks)

3

A bag contains only red, blue and green counters. A counter is taken from the bag at random. P(red) = 0.25 and P(blue) = 0.45.

(a)Work out P(green).(2)

(b)There are 40 counters in the bag. Work out the number of green counters.(2)

(Total for Question 3 is 4 marks)

4

Two fair coins are flipped at the same time.

(a)List all the possible outcomes.(2)

(b)Find the probability of getting at least one head.(2)

(Total for Question 4 is 4 marks)

5

Spinner A is equally likely to land on 1, 2 or 3. Spinner B is equally likely to land on 1 or 2. Both spinners are spun once and the two scores are added together.

(a)Complete a sample space diagram to show all the possible totals.(2)

(b)Find the probability that the total is greater than 3.(2)

(Total for Question 5 is 4 marks)

6

60 students were asked whether football or tennis is their favourite sport. The incomplete two-way table shows some of the results.

	Football	Tennis	Total
Boys	18	16	34
Girls	20	?	26
Total	?	22	60

(a)Complete the two-way table.(2)

(b)One student is picked at random from all 60 students. Find the probability that the student is a girl who prefers football.(2)

(Total for Question 6 is 4 marks)

7

80 adults were asked whether they own a car and whether they own a bike. 45 of the adults own a car. Of the adults who own a car, 18 also own a bike. Of the adults who do not own a car, 21 own a bike.

(a)Complete the frequency tree to show this information.(3)

(b)Work out the probability that a randomly selected adult owns a bike.(2)

(Total for Question 7 is 5 marks)

8

In a class of 30 students, 14 study French, 16 study Spanish and 5 study both French and Spanish. Every student studies French, Spanish, both, or neither.

(a)Work out the number of students who study only French.(1)

(b)Work out the number of students who study neither French nor Spanish.(2)

(c)A student is picked at random from the class. Find the probability that the student studies both French and Spanish. Give your answer as a fraction in its simplest form.(2)

(Total for Question 8 is 5 marks)

9

A biased coin has P(Heads) = 0.6. The coin is flipped twice. The tree diagram shows the possible outcomes, but some of the probabilities are missing.

(a)Write down the missing probabilities on the tree diagram.(2)

(b)Find the probability that the coin lands on heads both times.(2)

(c)Find the probability that the coin lands on exactly one head.(2)

(Total for Question 9 is 6 marks)

10

The probability that Tom passes his maths test is 0.7. The probability that he passes his science test is 0.8. The two events are independent.

(a)Work out the probability that Tom passes both tests.(2)

(b)Work out the probability that Tom fails both tests.(2)

(Total for Question 10 is 4 marks)

11

Priya rolls an ordinary six-sided dice 150 times. It lands on a six 40 times.

(a)Calculate the experimental probability of getting a six.(1)

(b)Priya says, 'this shows the dice is definitely biased.' Comment on whether Priya is correct.(2)

(Total for Question 11 is 3 marks)

12

The Venn diagram shows information about 50 members of a leisure centre. G = members who use the gym. S = members who swim. Only gym = 22, only swim = 14, both gym and swim = 9, neither = 5.

(a)Write down n(G ∩ S).(1)

(b)Write down P(G ∪ S).(1)

(c)Write down P(G').(2)

(Total for Question 12 is 4 marks)

13

A spinner can land on red, blue, green or yellow. The table shows the probability of each colour in terms of x.

Colour	Red	Blue	Green	Yellow
Probability	2x	3x	x	4x

(a)Work out the value of x.(2)

(b)Work out P(blue).(1)

(c)The spinner is spun 300 times. Work out the expected number of times it lands on green.(2)

(Total for Question 13 is 5 marks)

14

A bag contains only black, white and pink counters. One counter is taken from the bag at random. P(black) = 0.2, P(white) = 0.35, P(pink) = 0.45.

(a)Write down P(black or white).(1)

(b)There are 20 counters in the bag. Work out the number of pink counters.(2)

(Total for Question 14 is 3 marks)

15

Two fair, six-sided dice are rolled and the scores are added together.

(a)Show that there are 6 ways of getting a total of 7.(2)

(b)Find the probability that the total is at least 10.(2)

(Total for Question 15 is 4 marks)

16

The two-way table shows how 100 people travel to work.

(a)Write down the number of part-time workers who walk to work.(1)

(b)A person is picked at random from the 100 people. Find the probability that the person is full-time and travels by car.(2)

(c)Given that the person picked works full-time, find the probability that they travel by bus.(2)

(Total for Question 16 is 5 marks)

17

In a group of 90 students, P(studies Art) = 0.4, P(studies Music) = 0.3, and P(studies both Art and Music) = 0.1.

(a)Work out P(Art or Music).(2)

(b)Work out the number of students who study neither Art nor Music.(2)

(Total for Question 17 is 4 marks)

18

A box contains 5 red pens and 3 blue pens. Nadia takes a pen at random from the box and does not replace it. She then takes a second pen at random from the box.

(a)Complete the tree diagram to show the probabilities for both pens.(3)

(b)Work out the probability that Nadia takes two red pens.(2)

(c)Work out the probability that Nadia takes at least one blue pen.(3)

(Total for Question 18 is 8 marks)

19

A box contains 12 raffle tickets numbered 1 to 12. Two tickets are taken at random from the box, one after the other, without replacement.

(Total for Question 19 is 3 marks)

20

Bag A contains 4 red counters and 6 blue counters. Bag B contains 3 red counters and 7 blue counters. Aaron picks one of the two bags at random, then takes one counter at random from that bag.

(a)Complete the tree diagram to show this information.(3)

(b)Work out the probability that Aaron takes a red counter.(2)

(Total for Question 20 is 5 marks)

21

The probability that Elena is late for school on any given day is 0.15. Assume this probability is the same and independent each day.

(Total for Question 21 is 3 marks)

Mark scheme · 4.19 Probability

Question 1

(a) B1 B (0.15) selected
(a) Answer: B) 0.15
(b) B1 1/3 oe (accept 2/6, 0.333, 33.3%)
(b) Answer: 1/3
(c) B1 2/3 oe, ft: 1 - (b)
(c) Answer: 2/3

Question 2

(a) B1 0.7 oe (accept 7/10, 70%)
(a) Answer: 0.7
(b) M1 0.7 x 500 oe, ft their (a)
(b) A1 350 cao
(b) Answer: 350

Question 3

(a) M1 1 - 0.25 - 0.45 oe
(a) A1 0.3 cao
(a) Answer: 0.3
(b) M1 0.3 x 40 oe, ft their (a)
(b) A1 12 cao
(b) Answer: 12

Question 4

(a) M1 at least 3 correct outcomes shown
(a) A1 all 4 outcomes listed with no repeats: HH, HT, TH, TT
(a) Answer: HH, HT, TH, TT
(b) M1 identifies 3 outcomes with at least one head, ft their list
(b) A1 3/4 cao
(b) Answer: 3/4

Question 5

(a) M1 at least 4 of the 6 totals correct
(a) A1 all 6 totals correct: 2, 3, 3, 4, 4, 5
(a) Answer: Totals: 2, 3, 3, 4, 4, 5
(b) M1 identifies 3 outcomes greater than 3, ft their diagram
(b) A1 1/2 oe cao
(b) Answer: 1/2

Question 6

(a) B1 Girls Tennis = 6
(a) B1 Total Football = 38
(a) Answer: Girls Tennis = 6, Total Football = 38
(b) M1 20/60 oe
(b) A1 1/3 cao
(b) Answer: 1/3

Question 7

(a) B1 Car branch: Bike = 18, No bike = 27
(a) B1 No car branch: Bike = 21, No bike = 14
(a) B1 No car total = 35 shown correctly
(a) Answer: Car (45): Bike 18, No bike 27. No car (35): Bike 21, No bike 14.
(b) M1 18 + 21 oe, ft their tree
(b) A1 39/80 cao
(b) Answer: 39/80

Question 8

(a) B1 9 cao
(a) Answer: 9
(b) M1 30 - (9 + 11 + 5) oe
(b) A1 5 cao
(b) Answer: 5
(c) M1 5/30 oe
(c) A1 1/6 cao
(c) Answer: 1/6

Question 9

(a) B1 P(Tails) = 0.4 on the first branch
(a) B1 P(Tails) = 0.4 on both second-level branches
(a) Answer: 0.4 in each missing space
(b) M1 0.6 x 0.6
(b) A1 0.36 cao
(b) Answer: 0.36
(c) M1 (0.6 x 0.4) + (0.4 x 0.6) oe
(c) A1 0.48 cao
(c) Answer: 0.48

Question 10

(a) M1 0.7 x 0.8
(a) A1 0.56 cao
(a) Answer: 0.56
(b) M1 0.3 x 0.2
(b) A1 0.06 cao
(b) Answer: 0.06

Question 11

(a) B1 4/15 oe (accept awrt 0.27, 26.7%)
(a) Answer: 4/15
(b) C1 states or uses the theoretical probability 1/6 (awrt 0.167) for comparison
(b) C1 valid conclusion with reason, e.g. 'No - the experimental probability is higher than 1/6, but 150 trials may not be enough to be certain the dice is biased' oe
(b) Answer: No, not definitely - more trials would be needed to be sure

Question 12

(a) B1 9 cao
(a) Answer: 9
(b) B1 9/10 oe (accept 45/50)
(b) Answer: 9/10
(c) M1 14 + 5 oe (= 19, not in G)
(c) A1 19/50 cao
(c) Answer: 19/50

Question 13

(a) M1 2x + 3x + x + 4x = 1 oe
(a) A1 x = 0.1 cao
(a) Answer: x = 0.1
(b) B1 0.3 cao, ft their x
(b) Answer: 0.3
(c) M1 0.1 x 300 oe, ft their x
(c) A1 30 cao
(c) Answer: 30

Question 14

(a) B1 0.55 oe
(a) Answer: 0.55
(b) M1 0.45 x 20
(b) A1 9 cao
(b) Answer: 9

Question 15

(a) M1 at least 4 correct pairs listed, e.g. (1,6), (2,5), (3,4), (4,3)
(a) A1 all six pairs listed with no repeats: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) cso
(a) Answer: 6 ways (shown)
(b) M1 identifies 6 outcomes: (4,6), (5,5), (6,4), (5,6), (6,5), (6,6)
(b) A1 1/6 cao
(b) Answer: 1/6

Question 16

(a) B1 15 cao
(a) Answer: 15
(b) M1 38/100 oe
(b) A1 19/50 cao
(b) Answer: 19/50
(c) M1 12/60 oe
(c) A1 1/5 cao
(c) Answer: 1/5

Question 17

(a) M1 0.4 + 0.3 - 0.1 oe
(a) A1 0.6 cao
(a) Answer: 0.6
(b) M1 (1 - 0.6) x 90 oe, ft their (a)
(b) A1 36 cao
(b) Answer: 36

Question 18

(a) B1 first branches 5/8 and 3/8
(a) B1 second branches after Red: 4/7 and 3/7
(a) B1 second branches after Blue: 5/7 and 2/7
(a) Answer: First: 5/8, 3/8. After Red: 4/7, 3/7. After Blue: 5/7, 2/7.
(b) M1 5/8 x 4/7
(b) A1 5/14 cao (oe 20/56)
(b) Answer: 5/14
(c) M1 recognises P(at least one blue) = 1 - P(both red)
(c) dM1 1 - their (b) correctly using a common denominator
(c) A1 9/14 cao ft
(c) Answer: 9/14

Question 19

M1 3/12 oe, identifying 3 tickets greater than 9 out of 12
M1 3/12 x 2/11 oe
A1 6/132 = 1/22 cso (answer given, full correct working must be shown)
Answer: 1/22 (shown)

Question 20

(a) B1 first branches Bag A 0.5, Bag B 0.5
(a) B1 branches under Bag A: Red 0.4, Blue 0.6
(a) B1 branches under Bag B: Red 0.3, Blue 0.7
(a) Answer: 0.5/0.5 then 0.4/0.6 (Bag A) and 0.3/0.7 (Bag B)
(b) M1 (0.5 x 0.4) + (0.5 x 0.3)
(b) A1 0.35 cao
(b) Answer: 0.35

Question 21

P1 identifies strategy P(at least one late day) = 1 - P(no late days in 3 days)
M1 0.85³ oe (= 0.614125)
A1 0.385875 oe, awrt 0.386, cao
Answer: 0.385875 (awrt 0.386)

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