I have a time series of observations from a longitudinal study of individual objects. These observations are seen as discrete sequences of features, one sequence per object. The sequences have different lengths.
At each discrete point of time an individual sequence may turn out to be defective. An error on an individual sequence does not mean that the sequence terminates. It might be that other, correct observations might follow.
I am interested in the probability that a sequence is correct up to a certain length. For that reason I've modeled each sequence as a series of binary events where an error is a positive event.
The probability $P(E, S, L, T) = p$ of an error $E$ in the sequence $S$ of length $L$ at time $1 \le T \le L$ is independent from
- the point in time when the error occurs
- the point in time when the sequence starts
- other errors at the same individual sequence
- the length of the individual sequence
- other sequences
I want to estimate the probabilty that a sequence contains $n$ consecutive and correct observations starting from the first one.
Can this be modeled by the probability $P(E|T \gt n)$? How is this probability estimated? In a first approach using binomial distribution I would think that it is simply $(1-p)^n$. However, I'm wondering how to consider the different lengths of the sequences? Do I have to account for the length only if the probability of an error ist not independent of the point in time when the error occurs?
Note: In this special case it is possible to use survival analysis. However, I am interested in other ways to estimate the probability.
Edit The events of interest are errors in data acquisition and in the way how observations of the same objects are related to each other: The observations are made on a yearly base. The data are pseudonomized on the base of name, birthday etc. and a irreversible identifier is generated. After that all personal data are removed. If an error occurs then this has nothing to do with any retained property of the individual object. Therefore time is the only variable and as mentioned the error is independent of it.
Edit 2 As Gilbert note in his answer I am actually asking for the survival probability $S(n)$ while $P(E|T \gt n)$ is different. I reformulate my question as follows: "What is the logical formula for the probability of (a) a sequence of length greater than or equal to $n$, and (b1) no error occuring at all, or (b2) an error occuring after $n$?"