To find the probability that X happens (hard-drive failure, radioactive decay) within a certain time period, we integrate the pdf over the interval. For example, the probability that the failure happens

*after*a particular time t is:

(Naturally, since the cdf(t) is 1 minus this value).

Here is one form of Bayes theorem:

Now, consider two events E and F defined as follows:

Note that P(F and E) = P(F) because F is totally contained within E.

Then:

The probability of failure after time r + s, when we know that failure occurs after time r, does not depend on r at all but is only a function of s. This is the

*memoryless*property of the exponential function, alluded to in a previous post.

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