Is the gambler fallacy real, or does luck balance out?

Your friend is committing the classic gamblers fallacy. **The flaw in their reasoning:** "The probability of 11 heads in a row is $(0.5)^{11}$, which is extremely unlikely. So after 10 heads, the 11th being heads would be an extremely unlikely event." This is WRONG because the probability of 11 heads in a row IS indeed $(0.5)^{11}$, but we have ALREADY OBSERVED the first 10 heads. The conditional probability is: $$P(\text{11 heads} \mid \text{10 heads}) = \frac{P(\text{11 heads})}{P(\text{10 heads})} = \frac{(0.5)^{11}}{(0.5)^{10}} = 0.5$$ **Law of Large Numbers:** The LLN says that as $n \to \infty$, the proportion of heads approaches 50%. It does NOT say that deviations in one direction are "compensated" by deviations in the opposite direction. A run of 10 heads is not "balanced" by future tails — its balanced by the fact that 10 heads out of 10,000 flips is negligible. **Another way:** If you flip a coin 100 times and get 60 heads, the LLN doesnt predict 40 heads in the next 100 flips. It predicts 50 heads in the next 100, giving 110 heads out of 200 = 55%. The excess becomes proportionally smaller as $n$ increases.