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AQA GCSE · Question 05.1 · Probability
A school year has 78 students.
28 wear glasses.
1/4 of the students who wear glasses are left-handed.
30% of the students who do not wear glasses are left-handed.
ξ = students in the school year
G = wears glasses
L = left-handed
Complete the Venn diagram.
A school year has 78 students.
28 wear glasses.
1/4 of the students who wear glasses are left-handed.
30% of the students who do not wear glasses are left-handed.
ξ = students in the school year
G = wears glasses
L = left-handed
Complete the Venn diagram.
How to approach this question
Start by finding the number of students in the intersection (wear glasses AND are left-handed). Then find the number who only wear glasses. Next, calculate how many students do not wear glasses. From that group, find how many are left-handed. This gives you the "left-handed only" section. Finally, calculate the number of students who are in neither group and place this number outside the circles.
Full Answer
1. **Total students (ξ)** = 78
2. **Wear glasses (G)** = 28
3. **Wear glasses AND are left-handed (G ∩ L)**: 1/4 of 28 = 28 / 4 = 7
4. **Only wear glasses (G only)**: Total G - (G ∩ L) = 28 - 7 = 21
5. **Do not wear glasses**: Total students - Wear glasses = 78 - 28 = 50
6. **Do not wear glasses AND are left-handed (L only, but from the non-glasses group)**: 30% of 50 = (30/100) * 50 = 15
7. **Total left-handed (L)**: (G ∩ L) + (L only from non-glasses) = 7 + 15 = 22
8. **Only left-handed (L only)**: This is the 15 calculated in step 6.
9. **Neither wear glasses nor are left-handed**: Total students - (G only) - (G ∩ L) - (L only) = 78 - 21 - 7 - 15 = 35.
Alternatively: (Do not wear glasses) - (Do not wear glasses AND are left-handed) = 50 - 15 = 35.
**Completed Venn Diagram:**
- Region for G only: 21
- Intersection G ∩ L: 7
- Region for L only: 15
- Outside both circles: 35
This is a problem about organizing information into sets using a Venn diagram.
1. **Intersection (G and L):** We are told 1/4 of the 28 students who wear glasses are left-handed. So, the number in the intersection is (1/4) * 28 = 7.
2. **"G only" region:** There are 28 students in total in the G circle. Since 7 are in the intersection, the number who only wear glasses is 28 - 7 = 21.
3. **"L only" region:** First find the number of students who do not wear glasses: 78 - 28 = 50. We are told 30% of these are left-handed. So, 30% of 50 = 0.3 * 50 = 15. These 15 students are left-handed but do not wear glasses, so they go in the "L only" part of the diagram.
4. **Outside region (Neither G nor L):** The total number of students is 78. The number of students in any of the circle regions is 21 (G only) + 7 (both) + 15 (L only) = 43. So, the number outside the circles is 78 - 43 = 35.
Common mistakes
✗ Calculating 30% of 78 instead of 30% of the students who do not wear glasses.
✗ Placing 28 in the "G only" section instead of subtracting the intersection.
✗ Forgetting to calculate the number of students outside both circles.
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