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A group of 100 students were surveyed to find whether they drank tea (T), coffee (C) or a soft drink (D) at any time in the previous week - Junior Cycle Mathematics - Question 3 - 2014
Question 3
A group of 100 students were surveyed to find whether they drank tea (T), coffee (C) or a soft drink (D) at any time in the previous week. 24 had not drunk any of t... show full transcript
Worked Solution & Example Answer:A group of 100 students were surveyed to find whether they drank tea (T), coffee (C) or a soft drink (D) at any time in the previous week - Junior Cycle Mathematics - Question 3 - 2014
Step 1
Represent the above information on the Venn diagram.
Answer
To represent the information on a Venn diagram, we need to break down the given data into the various sets and intersections:
- Let’s denote:
- Let x be the number of students who only drank tea.
- Let y be the number of students who only drank coffee.
- Let z be the number of students who only drank a soft drink.
- Let p be the number of students who drank both tea and coffee only.
- Let q be the number of students who drank both tea and soft drinks only.
- Let r be the number of students who drank both coffee and soft drinks only.
- Let s be the number of students who drank all three.
From the problem,
- We know:
- Total students = 100
- Students who drank none = 24
- Students who drank tea = 41
- Students who drank either tea or coffee but not soft drinks = 51
- Students who drank at least two = 20
- Students who drank tea and soft drink but not coffee = 8
- Students who drank coffee and soft drink = 9
- Students who drank all three = 4
Setting up these equations, we find:
- From students who drank none:
- Total students = 100 - 24 = 76
- Using these variables in equations can help fill the Venn diagram:
- p + q + r + s = 20
- 8 + r + s = 41
- Hence, working through the remaining relations will help fill the parts of the Venn. This results in:
Final Venn Diagram:
- Tea (T): 26; Coffee (C): 22; Soft Drink (D): 8
- Intersection (TC): 4; (TS): 9; (CS): 3; (T only): 22; (C only): 8; (D only): 3.
Step 2
Find the probability that a student chosen at random from the group had drunk tea or coffee.
Answer
To find the probability that a student chosen at random had drunk tea or coffee, we first determine the number of students in the union of sets T and C:
Total number of students who drank tea or coffee = Students who drank only tea + Students who drank only coffee + Students who drank both tea and coffee + Students who drank all three.
- Hence, Total = 26 (only tea) + 22 (only coffee) + 4 (both tea and coffee) + 4 (all three) = 68.
The probability is then: egin{align*} P(T ext{ or } C) = rac{ ext{Number of students who drank tea or coffee}}{ ext{Total number of students}} = rac{68}{100} = 0.68. \end{align*}
Step 3
Find the probability that a student chosen at random from the group had drunk tea and coffee but not a soft drink.
Answer
To find the probability that a student chosen at random had drunk tea and coffee but not a soft drink:
- From our data, the number of students who drank tea and coffee but not soft drink is given as 4. Thus:
The probability is: egin{align*} P(T ext{ and } C ext{ not } D) = rac{ ext{Number of students who drank tea and coffee but not a soft drink}}{ ext{Total number of students}} = rac{4}{100} = 0.04.\end{align*}