Is there any general formula to calculate how many chances you have to win a particular poker hand?
Just outs to improve on the next card is fairly straight forward. Probability to win is much more complex. If it is just you against a single player on the river then you can calculate. A starting hand against 3 random hands is very complex.
I assume you mean chances (not change).
You have poker calculators but there is no formula
A straight can get beat by flush on the river that gets beat by a full house that gets beat by 4 of a kind. It is very complex to consider all the possibilities.
To run all the combinations is just a lot of runs. Over 4 players with today's computing power it is not even realistic.
For multiplayer they resort to statistical probability where millions of randoms hands are run.
You can use this this calculator, but basically you would need to know your opponents hand in order to actually calculate the odds.
The probability you have of hitting your outs however, can be calculated. Take a look at my other answer to learn how.
If you have a solid read you could try to include the probability of him hitting his hand after you have hit your hand. see this answer for an analysis. However, this would not be very practical at the poker table.