Every class in a school has 400 pupils. Of them, 50 study business studies, 300 study French, and 270 study math. Everybody who studies math also studies French. Twenty take business and math courses. 35 pursue business and French studies. A pupil is chosen at random. Calculate the likelihood that the chosen student studies business studies or French but not math or science. Only use Venn diagrams. [Answer Limit: 125 words] [UKPSC 2012]
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To solve this problem using a Venn diagram:
Define the Sets:
𝑀
M: Students studying Maths (270)
𝐹
F: Students studying French (300)
𝐵
B: Students studying Business Studies (50)
Given Data:
All Maths students also study French:
∣
𝑀
∩
𝐹
∣
=
270
∣M∩F∣=270
Students studying both Maths and Business Studies:
∣
𝑀
∩
𝐵
∣
=
20
∣M∩B∣=20
Students studying both French and Business Studies:
∣
𝐹
∩
𝐵
∣
=
35
∣F∩B∣=35
Calculate Intersections:
Students studying all three subjects:
∣
𝑀
∩
𝐹
∩
𝐵
∣
=
20
∣M∩F∩B∣=20
Students studying only French and Business Studies:
∣
𝐹
∩
𝐵
∣
−
∣
𝑀
∩
𝐹
∩
𝐵
∣
=
35
−
20
=
15
∣F∩B∣−∣M∩F∩B∣=35−20=15
Students studying only French:
Total studying French:
300
−
270
−
15
=
15
300−270−15=15
Probability:
Students studying only French = 15
Total students = 400
𝑃
(
only French
)
=
15
400
=
3
80
P(only French)=
400
15
=
80
3
Thus, the probability that the selected student studies French but neither Maths nor Business Studies is
3
80
80
3
.