[AaronBrown]

The ideas do not contradict because averages regress to the mean, totals wander without preferred direction. The Gambler's Fallacy is to apply regression to totals of independent events.

For example, after 280 rolls in Monopoly, the average square has been landed on 7 times (I'm ignoring some fine points about squares that direct you to other squares like Chance and Go To Jail, and rolls when you are in jail). Indiana has been landed on 0 times, which is also 0% of the rolls.

If you continue playing, the percentage of rolls resulting in someone landing on Indiana will tend to go up until it reaches its long-term average of 2.5% (again, ignoring the fine points that lead to unequal square probabilities). At that point, it will go up or down randomly, with equal probability. However, the total number of times someone has landed on Indiana, which is now 7 below its mean, has no tendency to go up or down (ignoring the fine point that you're more likely to land on Indiana if you know you aren't starting from Indiana). The expected deficit of Indiana landings after 1,000 or 100,000,000 rolls is 7.

This assumes that all moves are independent. There are things that regress toward the mean even in total. If interest rates are high relative to recent history they are more likely to go down than up. If you parent is taller than average, you are likely to be shorter than your parent (but taller than average). The baseball player with the highest batting average at the All-Star break is likely to have a lower average in the second half of the season than the first. These are all regressions different from the tendency of averages to move toward the long-term mean.