Q:

Guideline for rule-finding data analysis?

As a beginner user I have a lot of experience in ETL (extract, transform, load) process. I work with large datasets on daily basis. Data collecting is very time consuming and requires a lot of resources.
The problem arises when we have data but we don't know how to analyze and make it useful.
I have a list of numbers (range from 2 to 5) as a sample and I want to analyze it to discover rules how numbers act.
To make this analysis more simpler I will try to find a rule how number 5 is showing up. For example, there is sequence of numbers 3,4,4,2,4,3,2,4,3,5,3,2,4,3,4,2,4,5,2,4,3,3,4,5 etc. (I have the full sample of 1440 numbers for better results of analysis).
What data analysis technique is used to find out rules (or prediction model that tells when number 5 will show up)?
How results can be recorded?

A:

Your problem description is not clear or at least, I'm not sure if I understand clearly.  With that disclaimer, I suggest you look into two classes of predictive models:

Decision tree and Random forest: A quick search on decision tree and random forests will suggest you can you discover "rules" that tell you what will happen if this variable has value v1 and that variable has value v2, etc.. 
Second possibility is that your series looks like a time series problem. Don't get confused by word "time". If it's just a series of numbers/categories, it can potentially be a time series problem. Ex: temperature, share price variation, even language (sequence of words). 
You can use regular time series models like Arima or you can use deep learning models like RNN, LSTM - RNNs etc. 

You may want to learn more about basic concepts of machine learning including supervised learning, unsupervised learning, classification, regression, time series models etc.
Just one word: you may have to suitably prepare your data for any type models to work better. All the best. 

