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F08 Conditional Probability

EDEXCEL 1ST0 · Calculator allowed · about 75 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

A fair spinner has 8 equal sections, numbered 1 to 8, as shown. The spinner is spun once.

(a)List the possible outcomes where the number is greater than 5.(1)

(b)Given that the number is greater than 5, find the probability that the number is even.(2)

(Total for Question 1 is 3 marks)

A survey asked 60 pupils in Year 10 and Year 11 how they travel to school: by bus or on foot (walk). The two-way table shows the results.

	Bus	Walk	Total
Year 10	12	18	30
Year 11	20	10	30
Total	32	28	60

(a)Write down the number of Year 11 pupils who walk to school.(1)

(b)One pupil is chosen at random from the Year 10 pupils. Find the probability that this pupil travels by bus.(2)

(c)One pupil is chosen at random from those who walk to school. Find the probability that this pupil is in Year 10.(2)

(Total for Question 2 is 5 marks)

40 people were asked whether they prefer tea or coffee, and whether they take sugar in their drink. The two-way table shows the results.

	Sugar	No sugar	Total
Tea	8	12	20
Coffee	14	6	20
Total	22	18	40

(a)Write down the number of coffee drinkers who do not take sugar.(1)

(b)One person is chosen at random from the tea drinkers. Find the probability that this person takes sugar.(2)

(Total for Question 3 is 3 marks)

60 students were asked whether they own a cat and whether they own a dog. The two-way table shows the results.

	Owns a dog	Does not own a dog	Total
Owns a cat	9	21	30
Does not own a cat	15	15	30
Total	24	36	60

(a)Write down the number of students who own both a cat and a dog.(1)

(b)A student is chosen at random from those who own a cat. Find the probability that this student also owns a dog.(2)

(c)A student is chosen at random from those who do not own a dog. Find the probability that this student owns a cat.(2)

(Total for Question 4 is 5 marks)

Decide whether each statement is True or False, then answer part (iii).

(i)The notation P(A|B) is read as 'the probability of A given B'.(1)

(ii)If two events are independent, the outcome of one event changes the probability of the other.(1)

(iii)Which expression gives P(B|A)?(1)

A) P(A) x P(B)
B) P(A and B) / P(A)
C) P(A and B) / P(B)
D) P(A) / P(B)

(Total for Question 5 is 3 marks)

A bag contains 10 sweets: 6 red and 4 yellow. Chloe takes a sweet at random from the bag and eats it. She does not put the sweet back in the bag. The sweet Chloe took was red.

(a)Find the probability that the second sweet Chloe takes is also red.(2)

(b)Find the probability that the second sweet Chloe takes is yellow.(2)

(Total for Question 6 is 4 marks)

80 pupils were asked whether they have a packed lunch or a school dinner. The two-way table shows some of the results.

	Packed lunch	School dinner	Total
Boys	14		38
Girls		20	42
Total			80

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

(b)One pupil is chosen at random from the girls. Find the probability that this pupil has a school dinner.(2)

(c)One pupil is chosen at random from those who have a school dinner. Find the probability that this pupil is a boy.(2)

(Total for Question 7 is 6 marks)

Cards numbered 1 to 20 are placed in a bag. A card is drawn at random from the bag.

(a)List the multiples of 4 from 1 to 20.(1)

(b)Given that the number drawn is a multiple of 4, find the probability that it is also a multiple of 3.(2)

(Total for Question 8 is 3 marks)

For each situation described, state whether the two events are independent or dependent.

(a)Rolling a fair dice, and then rolling it again.(1)

(b)Taking two sweets at random from a bag, one after another, without replacing the first.(1)

(c)Explain, in the context of part (b), why the second event is dependent on the first.(2)

(Total for Question 9 is 4 marks)

The two-way table shows whether it rained on each of 90 days recorded during a statistics project, split by whether the day was a weekday or a weekend day.

	Rain	No rain	Total
Weekday	18	46	64
Weekend	8	18	26
Total	26	64	90

(a)Given that a day is a weekend day, find the probability that it rained.(2)

(b)Given that it rained, find the probability that the day was a weekday.(2)

(c)Using your answer to part (a), and a similar calculation for weekdays, comment on whether rain was more common on weekend days or on weekdays in this data set.(1)

(Total for Question 10 is 5 marks)

50 people were asked whether they wear glasses and whether they are aged over 40. The two-way table shows the results.

	Over 40	Under 40	Total
Glasses	18	7	25
No glasses	10	15	25
Total	28	22	50

One person is chosen at random from all 50 people.

(a)Write down P(Over 40 and Glasses).(1)

(b)Write down P(Over 40).(1)

(c)Use the formula P(Glasses | Over 40) = P(Over 40 and Glasses) / P(Over 40) to find P(Glasses | Over 40).(2)

(d)Show that your answer to part (c) is the same as the probability found by looking directly at the 'Over 40' column of the table.(1)

(Total for Question 11 is 5 marks)

A bag contains 9 counters: 5 blue and 4 green. Rhys takes a counter at random from the bag and does not replace it. He then takes a second counter at random from the bag. The tree diagram shows the first pick.

(a)Complete the tree diagram by writing the four missing second-pick probabilities.(2)

(b)Find the probability that both counters are blue.(2)

(c)Find the probability that the two counters are different colours.(2)

(Total for Question 12 is 6 marks)

200 students at a school travel to school either by bus or by car. Each student's journey on one morning was recorded as Late or On time. The completed frequency tree shows the results.

(a)A student is chosen at random from those who were late. Find the probability that this student travelled by car.(2)

(b)A student is chosen at random from those who travelled by bus. Find the probability that this student was on time.(2)

(c)State which method of transport, bus or car, had the higher proportion of on-time students, showing your reasoning.(1)

(Total for Question 13 is 5 marks)

A student carries out a statistics project to investigate whether pupils who play a musical instrument are more likely to also play a team sport.

(a)Write a suitable hypothesis for this investigation.(1)

(b)The student plans to collect data only from members of the school orchestra. Give one reason why this sampling method could make the investigation's conclusions unreliable.(1)

(c)The student instead surveys 90 pupils across the whole school. The two-way table shows the results.

	Plays a sport	Does not play a sport	Total
Plays an instrument	21	9	30
Does not play an instrument	24	36	60
Total	45	45	90

Find P(plays a sport | plays an instrument) and P(plays a sport | does not play an instrument).(3)

(d)Using your answers to part (c), comment on whether the data supports the hypothesis in part (a), and give one limitation of this conclusion.(2)

(Total for Question 14 is 7 marks)

For two events A and B, P(A) = 0.4, P(B) = 0.5 and P(A and B) = 0.2.

(a)Use the formula P(A|B) = P(A and B) / P(B) to find P(A|B).(2)

(b)Determine, giving a reason, whether events A and B are independent.(2)

(Total for Question 15 is 4 marks)

A box contains 12 cards, numbered 1 to 12. Priya takes a card at random from the box and does not replace it. She then takes a second card at random from the box.

(a)Given that the first card shows an even number, find the probability that the second card also shows an even number.(2)

(b)Find the probability that both cards show even numbers.(2)

(c)Find the probability that both cards show odd numbers.(2)

(Total for Question 16 is 6 marks)

A sports centre surveyed 120 members on whether they attend gym sessions and whether they attend swimming sessions. The two-way table shows the results.

	Gym	No gym	Total
Swim	27	21	48
No swim	33	39	72
Total	60	60	120

(a)Find P(Swim | Gym).(2)

(b)Find P(Swim).(2)

(c)Using your answers to parts (a) and (b), determine whether attending the gym and attending swimming sessions are independent events. Give a reason.(2)

(Total for Question 17 is 6 marks)

Mark scheme · F08 Conditional Probability

Question 1

(a) B1 6, 7, 8 (all three, any order)
(a) Answer: 6, 7, 8
(b) M1 restricts the sample space to {6, 7, 8}, oe
(b) A1 2/3 oe cao
(b) Answer: 2/3

Question 2

(a) B1 10 cao
(a) Answer: 10
(b) M1 12/30 seen, oe
(b) A1 2/5 oe cao
(b) Answer: 2/5
(c) M1 18/28 seen, oe
(c) A1 9/14 oe cao
(c) Answer: 9/14

Question 3

(a) B1 6 cao
(a) Answer: 6
(b) M1 8/20 seen, oe
(b) A1 2/5 oe cao
(b) Answer: 2/5

Question 4

(a) B1 9 cao
(a) Answer: 9
(b) M1 9/30 seen, oe
(b) A1 3/10 oe cao
(b) Answer: 3/10
(c) M1 21/36 seen, oe
(c) A1 7/12 oe cao
(c) Answer: 7/12

Question 5

(i) B1 True cao
(i) Answer: True
(ii) B1 False cao
(ii) Answer: False
(iii) B1 B cao
(iii) Answer: B) P(A and B) / P(A)

Question 6

(a) M1 5/9 seen, oe (5 red sweets remain out of 9 sweets left)
(a) A1 5/9 cao
(a) Answer: 5/9
(b) M1 4/9 seen, oe (4 yellow sweets remain out of 9 sweets left)
(b) A1 4/9 cao
(b) Answer: 4/9

Question 7

(a) B1 Boys school dinner = 24 and Girls packed lunch = 22
(a) B1 Total packed lunch = 36 and Total school dinner = 44
(a) Answer: Boys school dinner = 24, Girls packed lunch = 22, Total packed lunch = 36, Total school dinner = 44
(b) M1 20/42 seen, oe
(b) A1 10/21 oe cao
(b) Answer: 10/21
(c) M1 24/44 seen, using the total school dinner value from part (a)
(c) A1 6/11 oe cao
(c) Answer: 6/11

Question 8

(a) B1 4, 8, 12, 16, 20
(a) Answer: 4, 8, 12, 16, 20
(b) M1 restricts the sample space to {4, 8, 12, 16, 20} and identifies 12 as the only multiple of 3
(b) A1 1/5 oe cao
(b) Answer: 1/5

Question 9

(a) B1 Independent cao
(a) Answer: Independent
(b) B1 Dependent cao
(b) Answer: Dependent
(c) M1 identifies that the total number of sweets, and the number of each colour, changes after the first sweet is removed
(c) A1 correctly links this change to the probability of the second event depending on the outcome of the first
(c) Answer: Because removing the first sweet changes both the total number of sweets left and the number of each colour left, the probability of each outcome on the second pick depends on which sweet was taken first.

Question 10

(a) M1 8/26 seen, oe
(a) A1 4/13 oe cao
(a) Answer: 4/13
(b) M1 18/26 seen, oe
(b) A1 9/13 oe cao
(b) Answer: 9/13
(c) B1 correct comparison with supporting figures: P(rain | weekend) = 4/13 (approximately 0.31) is greater than P(rain | weekday) = 18/64 = 9/32 (approximately 0.28), so rain was slightly more common on weekend days
(c) Answer: Rain was slightly more common on weekend days: P(rain|weekend) = 4/13, approximately 0.31, compared with P(rain|weekday) = 9/32, approximately 0.28.

Question 11

(a) B1 18/50 oe (= 9/25)
(a) Answer: 18/50 (= 9/25)
(b) B1 28/50 oe (= 14/25)
(b) Answer: 28/50 (= 14/25)
(c) M1 (18/50) / (28/50), oe
(c) A1 9/14 oe cao
(c) Answer: 9/14
(d) B1 18/28 = 9/14, matching part (c)
(d) Answer: 18/28 = 9/14, which matches part (c).

Question 12

(a) B1 after Blue: P(Blue) = 4/8 and P(Green) = 4/8
(a) B1 after Green: P(Blue) = 5/8 and P(Green) = 3/8
(a) Answer: After first Blue: 4/8 Blue, 4/8 Green. After first Green: 5/8 Blue, 3/8 Green.
(b) M1 5/9 x 4/8, oe
(b) A1 5/18 oe cao
(b) Answer: 5/18
(c) M1 (5/9 x 4/8) + (4/9 x 5/8), oe
(c) A1 5/9 oe cao
(c) Answer: 5/9

Question 13

(a) M1 4/22 seen (total late = 18 + 4 = 22)
(a) A1 2/11 oe cao
(a) Answer: 2/11
(b) M1 102/120 seen, oe
(b) A1 17/20 oe cao
(b) Answer: 17/20
(c) B1 Car, supported by comparing 102/120 = 0.85 (bus) with 76/80 = 0.95 (car)
(c) Answer: Car

Question 14

(a) B1 a clear, testable hypothesis linking playing an instrument with playing a sport, e.g. 'Pupils who play a musical instrument are more likely to also play a team sport'
(a) Answer: Pupils who play a musical instrument are more likely to also play a team sport (accept any equivalent testable statement).
(b) B1 identifies bias, e.g. orchestra members already play an instrument so there is no fair comparison group, or the sample is too small/unrepresentative of the whole school
(b) Answer: Sampling only orchestra members is biased: it excludes pupils who do not play an instrument, so there is no fair comparison group and the sample may not represent the whole school.
(c) M1 21/30 and 24/60 identified
(c) A1 7/10 oe cao
(c) A1 2/5 oe cao
(c) Answer: P(sport | instrument) = 7/10; P(sport | no instrument) = 2/5
(d) B1 valid comparative comment supporting the hypothesis, e.g. 7/10 (0.7) is greater than 2/5 (0.4), so pupils who play an instrument are more likely to also play a sport
(d) B1 a valid limitation, e.g. the sample of 90 pupils from one school may not represent all pupils, or the data shows an association but does not prove that playing an instrument causes pupils to play sport
(d) Answer: The data supports the hypothesis, since 7/10 (0.7) is greater than 2/5 (0.4): pupils who play an instrument are more likely to also play a sport. Limitation: this is a sample of 90 pupils from one school, and an association does not prove that one causes the other.

Question 15

(a) M1 0.2 / 0.5, oe
(a) A1 0.4 cao
(a) Answer: 0.4
(b) M1 compares P(A|B) = 0.4 with P(A) = 0.4, oe compares P(A) x P(B) = 0.2 with P(A and B) = 0.2
(b) A1 correct conclusion: independent, since P(A|B) = P(A) (oe since P(A) x P(B) = P(A and B))
(b) Answer: Independent, since P(A|B) = 0.4 = P(A).

Question 16

(a) M1 identifies 6 even numbers in total, so 5 even numbers remain out of 11 cards
(a) A1 5/11 oe cao
(a) Answer: 5/11
(b) M1 6/12 x 5/11, oe
(b) A1 5/22 oe cao
(b) Answer: 5/22
(c) M1 6/12 x 5/11, oe (using the same reasoning as part (b), with 6 odd numbers in total)
(c) A1 5/22 oe cao
(c) Answer: 5/22

Question 17

(a) M1 27/60 seen, oe
(a) A1 9/20 oe cao
(a) Answer: 9/20
(b) M1 48/120 seen, oe
(b) A1 2/5 oe cao
(b) Answer: 2/5
(c) M1 compares 9/20 (= 0.45) with 2/5 (= 0.4)
(c) A1 correct conclusion: not independent, since P(Swim|Gym) does not equal P(Swim)
(c) Answer: Not independent, since P(Swim|Gym) = 9/20 does not equal P(Swim) = 2/5.

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