Is it possible to evaluate probability based on previous results? Let's consider a fair coin. I know that everytime this coin is tossed, the probability of getting a getting a head (H) is equal to the probably of getting a tail (T) which is 1/2 (or 50% / 50%)
Now, what if I toss the same coin N times and I record every outcome. In a scenario where I get H 999 times in a row, I would say, through common sense,  that the probability of getting a T is higher than the probability of getting a H(>1/2) in the 1000th tossing.
Is this common sense right? Can we really predict the probability of a random event based on its previous outcomes?
If yes, is there a mathematical model for this prediction?
 A: Every probabilistic model is always made for a specific purpose. The model that uses the assumption of a fair coin, i.e. all tosses are independent and $p=1/2$, is a model that may or may not be appropriate for a given experiment. If you want to do inference on $p$, it is clearly nonsensical to assume that the coin is fair. 
One of many probabilistic models for this scenario is to do a Bayesian analysis of the distribution of $p$ (the probability of heads). For example, let's assume every toss is independent but $p$ might be something different than $1/2$. Let $X$ be the number of heads. Then the probability of $n$ heads out of $N$ tosses can be modelled as a Binomial distribution:
$$
P(X=n)= \binom{N}{n}p^n (1-p)^{N-n}
$$
Now assume some prior beliefs about $p$. One possibility is that $p$ is Uniform in $[0, 1]$. Now, using Bayes' rule, the posterior distribution of $p$ becomes
$$
f(p \mid N=n) = \frac{P(N=n\mid p) f(p)}{P(N=n)} = \frac{\binom{N}{n}p^n(1-p)^{N-n}}{\int_0^1 \binom{N}{n}p^n(1-p)^{N-n} dp}
$$
In the case you have, where $n=N$, this expression becomes very simple:
$$
f(p \mid N=N) = (N+1)p^{N}
$$
So for example
$$
P(p > 1/2) = \int^1_{1/2} (N+1)p^{N} dp =   1-\frac{1}{2^{N+1}}
$$
Substituting $N = 999$ is clearly going to give a probability close to $1$ that this coin is loaded. 
As to whether the past can tell you something about the future, i.e. whether there is dependency between future and past realizations of a stochastic process, again that depends on your model assumptions. There is a myriad of stochastic processes which aim at incorporating such dependencies. One example is the Autoregressive Process which includes possible correlation between consecutive realizations. 
