Handling Data 4
7 marks · HardThe Venn diagram below shows the results of a survey of 100 students about whether they prefer football (F), hockey (H), or tennis (T).
The numbers indicate how many students are in each region. The '?' represents the number of students who did not prefer any of these sports.
Calculate the value of '?'.
Find the probability that a randomly selected student preferred at least two of the sports.
Given that a randomly selected student preferred hockey, find the probability that the student also preferred football.
A student is chosen at random. What is the probability that this student preferred exactly one sport?
Work this out on paper, then compare with the answer and worked solution below. With a free account you can submit written answers for marking.
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(a) 47
(b) 0.28
(c) 0.5
(d) 0.45
Must show: correct use of the Venn diagram numbers
Common mistakes: not including all regions for 'at least two'; not correctly summing regions of overlap
Show mark scheme and worked solution
Marking points
M1 (a) · 1 mark · Sums all the numbered regions and subtracts from 100 to find '?'.
M2 (b) · 1 mark · Identifies all regions with at least two sports (overlaps, including triple).
A1 (b) · 1 mark · Calculates correct probability (within tolerance). (follow-through)
M3 (c) · 1 mark · Finds total number preferring hockey; counts overlap with football.
A2 (c) · 1 mark · Correct conditional probability (within tolerance). (follow-through)
M4 (d) · 1 mark · Adds only the single-sport regions.
A3 (d) · 1 mark · Correct probability for exactly one sport (within tolerance). (follow-through)
Worked solution
a) Add all given numbers except '?': 20 (F only) + 6 (F & H only) + 5 (F & T only) + 8 (H only) + 4 (H & T only) + 7 (T only) + 3 (all three) = 53 So, '?' = 100 - 53 = 47.
b) At least two sports: overlap regions: 6 (F & H only) + 5 (F & T only) + 4 (H & T only) + 3 (all three) = 18 Probability = 18 / 100 = 0.18.
c) Total preferring Hockey: 8 (H only) + 6 (F & H only) + 4 (H & T only) + 3 (all three) = 21 Football and Hockey (i.e., F & H only + all three): 6 + 3 = 9 Conditional probability = 9 / 21 ≈ 0.429.
d) Exactly one sport: 20 (F only) + 8 (H only) + 7 (T only) = 35; probability = 35/100 = 0.35.