Why don't previous events affect the probability of (say) a coin showing tails? Why doesn't a previous event affect the probability of (say) a coin showing tails?
Let's say I have a fair and unbiased coin with two sides, heads and tails.
For the first time I toss it up the probabilities of both events are equal to $\frac{1}{2}$.
This much is intuitive and clear to me.
Now suppose that I toss it up $1000000000$ times and the scores are,
$501000000$ Heads
$499000000$ Tails
Now, for the $1001000000^{th}$ toss, shouldn't the probability of a tail coming up be greater than that of heads showing up?
I have seen many books which say that even for the $1001000000^{th}$ toss, the probabilities of both events are equal to $\frac{1}{2}$.
This seems wrong to me since the same books affirm that if a coin is tossed a large number of times, the quantity $\frac{heads}{tails}$ will approach $1$.
I know this is very elementary and naive, yet I had only superficially studied probability and I hope you all will bear with me.
My Objections with some of the top-voted answers

It isn't that future flips compensate for the imbalance, it is that there are so many of them it doesn't matter. 

I don't get this statement. What exactly does the second sentence mean? Moreover, if what you said is true then, the following comment by a user should be wrong,

Law of large numbers

So these are contradicting each other I feel. Please bear with my lack of knowledge.

The assumption for a coin is that is has no memory.  That means that the chance of heads is the same on every toss.  For a fair coin, that chance is $\frac 12$ regardless of the history.  If you toss $100$ times and get heads every time (very unlikely, but it could happen)  the most probable event after a million tosses (including the $100$ you already did) is $500050$ heads and $499950$ tails.  It isn't that future flips compensate for the imbalance, it is that there are so many of them it doesn't matter.  Look how close the head/tail ratio would be to $1$ at that point.
