# Probability change in successive coin tosses

Suppose I'm tossing an unbiased coin, and recording the outcomes. Let's say that by pure chance, I get 'n' consecutive heads. I'm confused about the probability of the (n+1)th coin toss.

When I asked my teacher this, she said that the probability of any (unbiased) coin toss will always be 1/2, regardless of the previous outcomes, as each toss is an independent event. This does make sense to me, but there's a further question.

Since the statistics have to balance out, i.e. after a considerable number of tosses the head-tail ratio should reach close to 1/2, isn't the probability of getting a tail also increasing, with each successive head that we get?

If not, and the probability of each coin toss is indeed 1/2, then what's changing as we keep getting heads?

Second, there are independent events and dependent events. Tossing a coin is independent of any past events. It's similar to picking up a ball each from $$n$$ different boxes with each box containing a black ball and a white ball. The probability of picking a white ball from $$5th$$ box is independent from the results from other boxes. It's always $$1\over2$$.
Now consider only one big box with $$2n$$ balls ($$n$$ white and $$n$$ black). You take a ball from the box and discard it outside. Now, the probability of getting white in $$5th$$ try would be dependent on previous outcomes.
If once you have obtained $$n$$ heads, and after that, you are going to bet for a consecutive head then you have a probability of $$1/2$$, the same you would get for whichever previous pattern.
The problem is completely different if, before starting the tosses, you bet that in the following $$n$$ tosses you are going to get $$m$$ heads consecutive. Then here you are going to consider the "statistics" of all the possible results in n tosses