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In a class of 30 students 17 study French (F) 20 study German (G) 2 do not study either subject - OCR - GCSE Maths - Question 11 - 2021 - Paper 1
Question 11
In a class of 30 students 17 study French (F) 20 study German (G) 2 do not study either subject. (a) Complete the Venn diagram. (b) Two of the 30 students are cho... show full transcript
Worked Solution & Example Answer:In a class of 30 students 17 study French (F) 20 study German (G) 2 do not study either subject - OCR - GCSE Maths - Question 11 - 2021 - Paper 1
Step 1
(a) Complete the Venn diagram
Answer
To complete the Venn diagram, we start by determining the number of students studying both French and German.
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Total Students: 30
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Students Not Studying: 2
Therefore, the number of students studying at least one subject is:
Total Students - Students Not Studying = 30 - 2 = 28
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Students Studying French (F): 17
Students Studying German (G): 20
To find the students studying both subjects, we can use the formula:
[ |F \cup G| = |F| + |G| - |F \cap G| ]
Where ( |F \cup G| ) = 28, ( |F| = 17 ), ( |G| = 20 ).
Thus,
[ 28 = 17 + 20 - |F \cap G| ]
Simplifying gives:
[ |F \cap G| = 9 ]
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Students Studying Only French:
[ 17 - |F \cap G| = 17 - 9 = 8 ] -
Students Studying Only German:
[ 20 - |F \cap G| = 20 - 9 = 11 ] -
Filling the Venn Diagram:
- French only: 8
- German only: 11
- Both French and German: 9
- Neither: 2
Thus, the completed counts are:
- French only: 8
- German only: 11
- Both: 9
- Neither: 2
Step 2
(b) Calculate the probability
Answer
To find the probability that one of the two students studies French but not German and the other studies German but not French, we need to analyze the two independent events:
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Students studying French but not German:
- From the Venn Diagram: 8 students.
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Students studying German but not French:
- From the Venn Diagram: 11 students.
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Total combinations of two students:
The total number of ways to pick 2 students from 30 can be calculated using the combination formula:[ \text{Total Combinations} = \binom{30}{2} = \frac{30 \cdot 29}{2} = 435 ]
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Favorable outcomes:
The number of favorable outcomes for choosing one student from each category:- 8 (French only) ( \times ) 11 (German only) = 88
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Calculating probability:
The probability of choosing one student from each category is given by:[ P = \frac{\text{Favorable outcomes}}{\text{Total combinations}} = \frac{88}{435} ]
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Conclusion: The probability that one student studies French but not German and the other studies German but not French is ( \frac{88}{435} ).