Gambler's fallacy and the Law of large numbers

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?)

• Perhaps you could explain more precisely where you think the contradiction lies; you'd be more likely to get responses that focus on your area of confusion. – rogerl Oct 13 '14 at 16:29
• i agree, but it's a bit hard to explain. i will edit my question however. – clamp Oct 13 '14 at 16:30
• Gamblers Fallacy has to do with conditional probability, as the Law of Large Numbers is an unconditional probability of outcome of average of large sample size. So law of large numbers says before you observe outcome of average the probability that it will equal the expected value will be close to one – Kamster Oct 13 '14 at 16:36
• For a fair coin, the probability of a million heads followed by one tail is the same as the probability of a million heads followed by another head. Namely $1/2^{1000001}$. A vague summary of the law of large numbers is not enough to tell you anything precise. – GEdgar Oct 4 '15 at 21:01
• Possible duplicate of Regression towards the mean v/s the Gambler's fallacy – quid Jul 20 '16 at 20:45

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?

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

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

• are we doing stats or probablity – Prince M Dec 4 '17 at 5:31
• I studied statistics and probability as a single discipline. The two are intermingled in my head. – James Webster Dec 4 '17 at 10:32

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.