Can someone explain me, how the Law of large numbers and the Gambler's Fallacy do not contradict.
The Gambler's Fallacy says, that there is no memory in randomness and any sequence of events has the same probability as any other sequence.
However, the Law of large numbers says, that given enough repetitions a certain event will likely happen.
To my understanding, these two kinda contradict each other because one says that you can not predict any random event but the other one says so (given enough repetitions of course).
For example imagine a series of coin tosses where the coin comes up heads a million times. The Gambler's fallacy says that the chance for the next toss to be tails is still 1/2. However the law of large numbers says, that since enough repetitions of tosses have come up heads, the next toss is more likely to be tails. (Which is definitely wrong?)
Best Answer
Any sequence has the same probability as any other, but there are more sequences that are "balanced" than any other given proportion. For example, if I flip a coin 4 times then there are 6 ways to get 2 heads and 2 tails. There's only one way to get all heads though.
The Gambler's Fallacy compares individual sequences (for instance, the sequences HHHHH and HHHHT).
The LLN talks about groups of sequences - it says which groups your result is more likely to fall into.