# How to estimate the probability of a rare event about which observations can only be made in quantized time?

I have reduced the estimation problem of a real event (a technical failure happening, the fact of which is checked in regular time intervals) to the following problem:

we have a non-fair coin, which gives a head for almost every throw. It gives a tail extremely rarely. Let's say we throw the coin every second. After an hour, a tail comes up. After two minutes, another. After 2 hours, another. How can we estimate the probability of a tail occurring, and how can we have a good guess at the reliability of the estimate after a given number of (e.g. few dozen) occurrences of a tail?

My problem is that the event is rare enough that it's very hard to get a reliable measurement for some of the longest time periods when it doesn't occur (it just takes a lot of time).

• What about a simple exponential distribution for the time between failures? Commented Nov 28, 2013 at 13:15
• Have you looked into survival analysis? Commented Nov 28, 2013 at 15:27