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.

Even in the 1000th time with a 999 tails, 0 heads, the probability of head is 1/2. But the probability that you get 1000th times tails in a row is 0.5^1000,  9.332636e-302.
