Question

A computer program was tested 300 times before its release. The testing was done in three stages of 100 tests each. The software failed 15 times in Stage I, 12 times in Stage II, 8 times in Stage III, 6 times in both Stage I and Stage II, 7 times in both Stage II and Stage III, 4 times in both Stage I and Stage III, and 4 times in all the three stages. How many times the software failed in a single stage only?

• A. 10
• B. 13
• C. 15
• D. 17
• E. 35

Hint:

When pairwise counts include the triple overlap, use \(|A\text{ only}|=|A|-|A\cap B|-|A\cap C|+|A\cap B\cap C|\).
Draw a Venn diagram to avoid double subtraction mistakes.

Answer 1

The Correct Option is B

Step 1: Let \(I, II, III\) be the sets of tests where the software failed in Stage I, Stage II, and Stage III respectively. Given: \(|I|=15,\ |II|=12,\ |III|=8\); pairwise intersections \(|I\cap II|=6,\ |II\cap III|=7,\ |I\cap III|=4\); triple intersection \(|I\cap II\cap III|=4\).
[2mm] Step 2: Failures in exactly one stage:
\[ \begin{aligned} |I\text{ only}| &= |I|-|I\cap II|-|I\cap III|+|I\cap II\cap III|=15-6-4+4=9,
|II\text{ only}|&=|II|-|I\cap II|-|II\cap III|+|I\cap II\cap III|=12-6-7+4=3,
|III\text{ only}|&=|III|-|I\cap III|-|II\cap III|+|I\cap II\cap III|=8-4-7+4=1. \end{aligned} \] Step 3: Total failures in a single stage only \[ 9+3+1=13. \]