A software security audit evaluated 500 open-source repositories for three specific vulner… — GRE General Test

A software security audit evaluated $500$ open-source repositories for three specific vulnerability types: SQL Injection ( $S$ ), Cross-Site Scripting ( $X$ ), and Buffer Overflow ( $B$ ). Exactly $80$ repositories had none of these vulnerabilities. The audit revealed the following data:

- $220$ repositories contained SQL Injection vulnerabilities.
- $190$ repositories contained Cross-Site Scripting vulnerabilities.
- $210$ repositories contained Buffer Overflow vulnerabilities.
- $75$ repositories contained both SQL Injection and Cross-Site Scripting vulnerabilities.
- $80$ repositories contained both Cross-Site Scripting and Buffer Overflow vulnerabilities.
- $30$ repositories contained all three vulnerability types.

How many of the audited repositories contained Buffer Overflow vulnerabilities ONLY?

A
$55$
$85$ Answer
C
$100$
D
$130$
E
$155$

Answer

85 repositories contained Buffer Overflow vulnerabilities only.

The total number of repositories containing at least one vulnerability is

500 - 80 = 420

. Using the inclusion-exclusion formula

|S \cup X \cup B| = |S| + |X| + |B| - (|S \cap X| + |X \cap B| + |S \cap B|) + |S \cap X \cap B|

, we substitute the known values:

420 = 220 + 190 + 210 - (75 + 80 + |S \cap B|) + 30

. Solving gives

420 = 650 - 155 - |S \cap B|

, so

|S \cap B| = 75

. The repositories containing Buffer Overflow ONLY are given by subtracting the overlapping regions from the total Buffer Overflow set:

|B| - (|S \cap B| \text{ only}) - (|X \cap B| \text{ only}) - |S \cap X \cap B| = 210 - (75 - 30) - (80 - 30) - 30 = 210 - 45 - 50 - 30 = 85

Step-by-Step Solution

Calculate the total number of repositories containing at least one vulnerability

|S \cup X \cup B| = 500 - 80 = 420

The total population is 500 and 80 repositories have no vulnerabilities.

Apply the 3-Set Inclusion-Exclusion Principle to determine the sum of pairwise intersections

|S \cap X| + |X \cap B| + |S \cap B| = 230

From

|S \cup X \cup B| = |S| + |X| + |B| - (|S \cap X| + |X \cap B| + |S \cap B|) + |S \cap X \cap B|

, we get

420 = 220 + 190 + 210 - \sum |\text{intersections}| + 30

, so

\sum |\text{intersections}| = 650 - 420 = 230

Find the missing pairwise intersection

|S \cap B|

|S \cap B| = 75

Since

|S \cap X| = 75

and

|X \cap B| = 80

, we have

75 + 80 + |S \cap B| = 230

, which yields

|S \cap B| = 75

Calculate the number of repositories with Buffer Overflow ONLY

|B \text{ only}| = 85

Subtract the exclusive two-set overlaps and the three-set overlap from

|B|

|B \text{ only}| = |B| - (|S \cap B| - |S \cap X \cap B|) - (|X \cap B| - |S \cap X \cap B|) - |S \cap X \cap B| = 210 - (75 - 30) - (80 - 30) - 30 = 210 - 45 - 50 - 30 = 85

Key Concept

Three-Set Inclusion-Exclusion Principle and Venn Diagram Region Analysis

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