Contradiction between the Law of Large Numbers and the Gambler's Fallacy? If I flip a fair coin 10 times, and all of them land on heads, according to the Law of Large Numbers I am more likely to get tails on the next flip. However, this is clearly the Gambler's Fallacy. 
How am I misinterpreting the Law of Large Numbers?
 A: Don't worry, this is a common misunderstanding. Let's consider this. 

If I flip a fair coin 10 times, and all of them land on heads,
  according to the Law of Large Numbers I am more likely to get tails on
  the next flip. However, this is clearly the Gambler's Fallacy. 
How am I misinterpreting the Law of Large Numbers?

The part I bolded is where the issue is, since the follow-up - that this is clearly the Gambler's Fallacy - is the correct angle. 
The Law of Large Numbers states that 

the average of the results obtained from a large number of trials
  should be close to the expected value, and will tend to become closer
  as more trials are performed.

The first issue is kind of a footnote - that ten trials isn't near enough to qualify for the "large number." (How large the large number has to be depends on the distribution - in the case of the binomial coin toss, it's roughly 30.) 
There is a deeper issue, though - even if you got the coin to land on heads one million times in a row, the odds of the next toss are still 50/50. How do we reconcile this? 
The Law of Large Numbers' stock-in-trade is in ratios. Over huge numbers, the ratio of "interesting events::possible outcomes" approximates the probability of an interesting event happening on any single occasion. For a concrete example - it's actually plausible that you flip a coin one million times as heads more than tails in the long run, where your ratio is something like 100.1:100 million (heads:tails). The "Big Idea" here is that, when you run a huge number of trials, that large number in the denominator will dilute the freak ten-heads streaks that appear and otherwise substantial numbers (say, 1,000,000) on either side of the scale matter a lot less to the ratio between them. 
(For further reading, Jordan Ellenberg does a fantastic job explaining this early in his book, How Not to be Wrong.)
