If a piece of computer equipment will fail once a year and be unavailable for 1.5 hours, how do I calculate the probability of the failure occurring each hour of every day
If your question is:
"What is the probability of a failure occurring in a given hour?"
We will count the failure event itself, which happend once per year, giving:
Number of hours in a year = 365d * 24h = 8,760h
Number of failures per hour = 1 failure per year / 8,760h/y = 0.0001142 failures per hour, or:
0.01142% chance of experiencing a failure in a given hour.
If the question was:
"What is the probability of having the computer be unavailable within a given hour?"
(as is implied by the duration given), then:
The probability of the single failure overlapping with a given hour is:
Number of days in a year = 365d/y
Number of days of unavailability = 1.5h / 24h/d = 0.0625d of unavailability per year
0.01712% probability of having some unavailability within a given hour.