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 p. Then 10p = _______.

GATE 2014 · Engineering Mathematics · Probability · medium

Answer: 10p = 25/21 ≈ 1.19, so 100p ≈ 11.90 (official answer ~11.9 means the quantity asked is 100p or equivalently p ≈ 0.1190)

1. Count total inspection outcomes: C(10, 4) = 10! / (4! * 6!) = 210.
2. Count outcomes with exactly 3 working: C(4,3) * C(6,1) = 4 * 6 = 24.
3. Count outcomes with exactly 4 working: C(4,4) * C(6,0) = 1 * 1 = 1.
4. Compute p and then 10p: p = (24 + 1) / 210 = 25/210 = 5/42. Then 10p = 50/42 = 25/21 ≈ 1.190.