Question 1185243: 1. A pupil attending university for the first time examined the previous year’s
statistics. He saw that 70% of the students passed mathematics, 30% passed
chemistry and 80% passed at least one of the two subjects. The pupil takes both maths and chemistry. If x=probability that the pupil takes both the subject, draw a Venn diagram to determine the percentage of passing
a) Both
b) Just one subject
c) Neither.
Answer by ikleyn(52921)   (Show Source):
You can put this solution on YOUR website! .
Venn diagram will represent one rectangle and two intersecting circles inside the rectangle.
One circle (M) will represent 70% of the total area of the rectangle.
Another circle (C) will represent 30% of the total area of the same rectangle.
The union of the two circles (M U C) will represent 80% of the area of the rectangle
(those students who passed at least one of the two exams).
The intersection (M ∩ C) represents those students who passed both exams.
For these quantities, n(M), n(C), n(M U C) and n(M ∩ C), the formula is valid
n(M U C) = n(M) + n(C) - n(M ∩ C),
or, in terms of percentage
80% = 70% + 30% - n(M ∩ C).
From the last equation, you get
n(M ∩ C) = 70% + 30% - 80% = 20%.
Thus 20% of the pupils passed both exams.
70% - 20% = 50% passed Math only.
30% - 20% = 10% passed Chemistry only.
In total, 50% + 10% = 60% passed one exam, only (one of the two exams).
And, finally, 100% - 80% = 20% is the percentage of those pupils who passed neither exam.
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Solved.
I answered all questions of the problem, and explained all the " why ? " to you.
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If you want to learn the subject wider, deeper and better, look into my lesson
- Counting elements in sub-sets of a given finite set
in this site.
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