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GATE CS 2014 Set 2 · Engineering Mathematics

Question 11

NATPrinted as CS Q.11 markNo negative markingMedium (estimate)

Question

The security system at an IT office is composed of 10 computers of which exactly four are working. To check whether the system is functional, the officials inspect four of the computers picked at random (without replacement). The system is deemed functional if at least three of the four computers inspected are working. Let the probability that the system is deemed functional be denoted by pp. Then 100p=100p = _____.

Official answer and solution
Accepted answers
11.85 to 11.95
From
Official GATE 2014 answer key

Solution

Model the sampling without replacement using the hypergeometric distribution for at least 3 working computers.

  1. Problem Parameters:

    • Total computers N=10N = 10.
    • Working computers K=4K = 4.
    • Defective computers N−K=6N - K = 6.
    • Sample size n=4n = 4 inspected without replacement.

  2. Total Possible Outcomes:

    (104)=10×9×8×74×3×2×1=210.\binom{10}{4} = \frac{10 \times 9 \times 8 \times 7}{4 \times 3 \times 2 \times 1} = 210.

  3. Favorable Outcomes (at least 3 working computers):

    • Case 1 (exactly 3 working): Choose 3 working and 1 defective:
      (43)×(61)=4×6=24.\binom{4}{3} \times \binom{6}{1} = 4 \times 6 = 24.
    • Case 2 (exactly 4 working): Choose 4 working and 0 defective:
      (44)×(60)=1×1=1.\binom{4}{4} \times \binom{6}{0} = 1 \times 1 = 1.

    Total favorable outcomes = 24+1=2524 + 1 = 25.

  4. Probability pp and Scaled Value 100p100p:

    p=25210=542≈0.1190476.p = \frac{25}{210} = \frac{5}{42} \approx 0.1190476.
    100p=50042≈11.9048.100p = \frac{500}{42} \approx 11.9048.

This falls within the official answer key range [11.85,11.95][11.85, 11.95].

Final answer

11.9

Common trap

Assuming sampling with replacement, which would incorrectly yield a binomial distribution.

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper
GATE 2014 Computer Science and Information Technology (CS) Session 2
Question
11 of 65, printed as CS Q.1
Section
Computer Science and Information Technology

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