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Difficulty: MediumConditional Probability

A software company audited 120120 applications for compliance with two protocols: Accessibility (Protocol A) and Security (Protocol B). The audit revealed that 7575 applications complied with Protocol A, 6060 applications complied with Protocol B, and 3535 applications complied with neither protocol. If an application selected at random from the audited group is known to comply with Protocol A, what is the probability that it also complies with Protocol B?

  1. A
    512\frac{5}{12}
  2. 23\frac{2}{3}Answer
  3. C
    56\frac{5}{6}
  4. D
    215\frac{2}{15}
  5. E
    1724\frac{17}{24}

Answer

23\frac{2}{3}
Out of the 120 total applications, 35 comply with neither protocol, leaving 85 that comply with at least one. Using the inclusion-exclusion principle (75+60−85=5075 + 60 - 85 = 50), exactly 50 applications comply with both protocols. Since the application is already known to comply with Protocol A, the denominator is restricted to the 75 applications in Protocol A. The conditional probability is therefore 50/75=2/350 / 75 = 2/3.

Step-by-Step Solution

1
Determine the number of applications that comply with at least one protocol.
Total applications minus those complying with neither: 120−35=85120 - 35 = 85.
The total sample space is partitioned into applications complying with at least one protocol and those complying with neither.
2
Find the number of applications complying with both Protocol A and Protocol B using the Principle of Inclusion-Exclusion.
∣A∩B∣=∣A∣+∣B∣−∣A∪B∣=75+60−85=50|A \cap B| = |A| + |B| - |A \cup B| = 75 + 60 - 85 = 50.
Summing the counts of Protocol A and Protocol B counts applications in both protocols twice.
3
Calculate the conditional probability P(B∣A)P(B|A).
P(B∣A)=∣A∩B∣∣A∣=5075=23P(B|A) = \frac{|A \cap B|}{|A|} = \frac{50}{75} = \frac{2}{3}.
Given that the selected application complies with Protocol A, the sample space is restricted to ∣A∣=75|A| = 75.

Key Concept

Conditional Probability with Overlapping Sets
Estimated Time:1m 30s
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Topics

Conditional Probability

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