I understand that each toss has a 50% chance if it is a fair coin, but I have hard time grasping the law of great numbers and I would like to know how likely it is that I get 20 heads in a row in such a large number of tosses.
Btw, is there really any difference if the coin IS NOT fair? Lets say that I only have a 47.5% chance of winning, does that increase the probability of getting 20 heads in a row?
Regards,
Hint for your first question:
Write $0$ if tais and $1$ if heads. Then the results of the 100-million coin tosses can be represented as a bitstring.
The probability of all possible 100-million digit bitstrings appearing are equal, since each coin flip is fair.
So you only have to count the number of 100-million digit bitstrings with at least one 20 digit subsequence of purely $1$s.
Edit: It turns out that you can use this recursive formula to calculate the probability of a run of K events in a row at least once in a series of N trials. In the final step of my hint, one may even have to resort to this formula in the end.