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Zorluk: ZorOverlapping Sets and Venn Diagrams

At a commercial bank, an audit of 300300 business loan applications revealed that each application underwent at least one of three specialized risk evaluations: Credit Risk, Market Risk, and Operational Risk. Exactly 180180 applications underwent Credit Risk evaluation, 150150 underwent Market Risk evaluation, and 135135 underwent Operational Risk evaluation. If exactly 9595 applications underwent exactly two of the three risk evaluations, how many loan applications underwent all three risk evaluations?

Cevap: 35 applications

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35 loan applications underwent all three risk evaluations.
The total number of applications is 300300. The sum of individual evaluations is 180+150+135=465180 + 150 + 135 = 465. Using the Venn diagram region formula, Sum of individual sets−Total=Exactly 2+2×(Exactly 3)\text{Sum of individual sets} - \text{Total} = \text{Exactly } 2 + 2 \times (\text{Exactly } 3). Substituting the known values gives 465−300=95+2x465 - 300 = 95 + 2x, which simplifies to 165=95+2x165 = 95 + 2x. Solving for xx gives 2x=702x = 70, so x=35x = 35.

Adım Adım Çözüm

1
State the fundamental overlapping set region equation for 3 sets.
\text{Total} = \text{Exactly } 1 + \text{Exactly } 2 + \text{Exactly } 3 + \text{Neither}
Every application falls into exactly one of these non-overlapping region categories.
2
Calculate the sum of the individual set counts.
180 + 150 + 135 = 465
Summing individual set counts counts items in 1 set once, 2 sets twice, and 3 sets three times.
3
Formulate the algebraic identity linking individual sums to region totals.
\text{Sum of Individual Sets} = \text{Exactly } 1 + 2(\text{Exactly } 2) + 3(\text{Exactly } 3)
This accounts for the multiple counting of overlapping regions.
4
Subtract the Total equation from the Sum equation to eliminate the 'Exactly 1' term.
465 - 300 = \text{Exactly } 2 + 2(\text{Exactly } 3) \Rightarrow 165 = 95 + 2(\text{Exactly } 3)
Subtracting (Exactly 1+Exactly 2+Exactly 3)(\text{Exactly } 1 + \text{Exactly } 2 + \text{Exactly } 3) from (Exactly 1+2⋅Exactly 2+3⋅Exactly 3)(\text{Exactly } 1 + 2\cdot\text{Exactly } 2 + 3\cdot\text{Exactly } 3) leaves 1⋅Exactly 2+2⋅Exactly 31\cdot\text{Exactly } 2 + 2\cdot\text{Exactly } 3.
5
Solve the linear equation for the 'Exactly 3' intersection.
2(\text{Exactly } 3) = 165 - 95 = 70 \Rightarrow \text{Exactly } 3 = 35
Dividing the remaining difference of 70 by 2 gives the number of applications in all three sets.

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Three-Set Venn Diagram Region Decomposition
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Overlapping Sets and Venn Diagrams

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