Suppose one wants to detect "bad" or "suspicious" subsequences in event sequences. Whether an observed pattern of activity (in the form of a particular subsequence) is significant and should be a cause for alarm, depends on how likely it is to occur fortuitously. A long enough sequence of observed events will almost certainly contain any subsequence, and setting thresholds for alarm is an important issue in a monitoring system that seeks to avoid false alarms. Suppose a long sequence T of observed events contains a suspicious subsequence pattern S within it, where the suspicious subsequence S consists of m events and spans a window of size w within T. We address the fundamental problem: is a certain number of occurrences of a particular subsequence unlikely to be fortuitous (i.e., indicative of suspicious activity)? If the probability of fortuitous occurrences is high and an automated monitoring system flags it as suspicious anyway, then such a system will suffer from generating too many false alarms. This paper quantifies the probability of such an S occuring in T within a window of size w, the number of distinct windows containing S as a subsequence, the expected number of such occurrences, its variance, and establishes its limiting distribution that allows to set up an alarm threshold so that the probability of false alarms is very small. We report on experiments confirming the theory and showing that we can detect bad subsequences with low false alarm rate.