Gambler's fallacy and the Law of large numbers

Solution 1:

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.

Solution 2:

The keyword here for me is given.

...that given enough repetitions...

Your Gambler's Fallacy quote mentions that there is no memory in randomness, which is true, the events are independent. But the use of the word "given" introduces memory and future events become "dependant".

If I ask you:

What is the chance of flipping two heads (of a fair coin) in a row?

Your calculation would be:

$$ P(A \& B) = P(A) \times P(B) ={1\over 2} \times {1\over 2} = {1\over 4} $$

If I ask you:

Given that I just flipped heads, what is the chance of flipping two heads in a row?

Your calculation is:

$$ P(A | B) = { P(A \& B) \over P(A) } = {{1\over 2} \times {1\over 2} \over {1 \over 2}} = {1 \over 2} $$

_{Would be good if somebody could check my maths. It's been about 10 years since I did stats!}