Pseudo-Random Binary Sequence Prediction

Cryptographically secure pseudorandom number generators are intended specifically to make what you want to do impossible. In particular, they satisfy the "next bit test": given k bits of their output, you cannot guess bit k+1 with probability greater than 1/2.

Plain pseudorandom number generators that do not satisfy the next bit test can be attacked and in fact security vulnerabilities have been discovered in real world systems due to the choice of PRNG. In particular, linear congruential generators are known to be somewhat (or completely) predictable, and some versions of Unix random may use this algorithm. This method is quite math intensive though. If you want to go down this path a search for "linear congruential generator prediction" is a place to start.

Another attack if you are aware of the PRNG implementation is to try to determine the seed used to generate the sequence you are analyzing. The seed is sometimes based on guessable information like time of day, process ID, etc.

Well, for pseudo-random sequences, the only possibility is to keep count how many of each possibility has come before. If the 1s outweigh the 0s, it's more likely that the next one will be 0. How much more likely depends on the relative occurrences of each.

Note that this won't work for true randomness since the events are independent, despite what the statisticians tell you :-)

You'll find that out (painfully) the first time you get a run of 13 reds on the table when you're using the double-on-loss method of playing roulette. In any case, the house derives its advantage from 0 (and double-0 on some tables) which are neither red nor black.

This is a decent question but I think if "you can barely play solitaire" it might be out of your reach right now.

You should look into picking up a basic language, and most are going to say PHP but I'm wary of recommending that to a beginner (it's pretty easy to get working though, see:XAMPP). Java is probably an "easy-to-get-running-and-work-with" language but I'm sure there's better threads on here about which language to start with (Python or something probably wins because experienced programmers love it).

By the way, your English is fine (I didn't notice you were a non-native English speaker).

Now, as for your question, if you're looking at true pattern matching. I'd be inclined to convert this idea to code:

 "CURRENTPOINT" is end of first letter.
 LOOP: Pick letter(s) from Start to "CURRENTPOINT"
 Break the rest of your binary string into blocks of the same size.
 See if these blocks all equal your picked letters.
 If not, move "CURRENTPOINT" along and repeat the LOOP until you run out of letters.
 If so, you have your "repeating section."

If you're just guessing that the random generator is temporarily biased, and that this bias will re-establish a baseline (balanced 0s and 1s) in the reasonably short-term then you can compare the count of each 0s and 1s and say the other is more likely based on the deviation from your baseline. However, be careful of the Monte Carlo fallacy.

To answer the PS first: No, because roulette spins are independent events so there's nothing predictive in the historical sequence of outcomes.

The general question is hard and interesting. This website can infer a surprising number of sequences from their initial values:

The On-Line Encyclopedia of Integer Sequences (OEIS)

Note that it's for arbitrary integer sequences.

I tried it on simple patterns like {0,0,1,1,0,0,1,1,...} and it says the right thing.

I noticed that nobody told you about periodicity.

Pseudo-random sequence always works on mathematical operation. (until the quantic computer ^^)

An usual way to generate one is to divide two prime number (not sure it's the right word but whatever).

for instance

1/3=1.333333.....
9/7=1,2857142857142857142857142857143

Those are fairly small number and what do we notice? Periodicity.

1/3=1.3 3 3 3 3 3.....
9/7=1,2857 142857 142857 142857 142857 143

The more big is the prime number the more the sequence in that case: 3 and 142857 will be big

So if you look to a pseudo-random sequence for a long time you may find a periodicity and be able to "guess" the next number. But that could take a while.

PS: sorry for my English, I’m a bit rusty ^^

What you need to think about is the properties of randomness, study those. For example, "Randomness runs in bunches". Compare a random sequence against a predictable sequence: you won't normally find bunches in the predictable one. To take advantage of bunches wait for the bunch. And with a little luck you will win.