An experimental file server is up 75% of the time and down for 25% of the time…

Answer: 4 servers.

Rationale: If each replica is up 75% of the time, the probability a single replica is down is 0.25. For n independent replicas, the system is unavailable only if all replicas are down, so combined availability = 1 - (0.25)^n.

• Set the requirement: 1 - (0.25)^n ≥ 0.99, which is equivalent to (0.25)^n ≤ 0.01.

• Take natural logs: n · ln(0.25) ≤ ln(0.01). Therefore n ≥ ln(0.01) / ln(0.25) ≈ 3.3219.

• n must be an integer, so the smallest integer ≥ 3.3219 is 4.

Quick check: For n = 4, availability = 1 - (0.25)^4 = 1 - 0.00390625 = 0.99609375 ≥ 0.99. For n = 3, availability = 1 - (0.25)^3 = 0.984375 < 0.99, so three replicas are insufficient.

Conclusion: You need 4 replicas to achieve at least 99% availability.