The law of large numbers is a fundamental concept in probability theory and statistics. In simple terms, it states that as you increase the number of trials or events, the average outcome will converge towards the expected value.

Let’s use a simple example – flipping a fair coin. We know each flip has a 50/50 chance of landing on heads or tails. That’s the expected value or probability. Now, if you flip the coin just a few times, you might get all heads or all tails by chance. The small sample size allows for random fluctuations and streaks.

However, as you keep flipping the coin hundreds or thousands of times, the ratio of heads to tails will approach the 50/50 expectation. A large enough sample size drowns out the random noise and reveals the underlying probability. That’s the law of large numbers in action.

Casinos rely heavily on the law of large numbers to maintain their edge over players. While a player might go on a hot streak and win a lot in the short term, the casino knows that over a longer time period, the odds are stacked in their favor.

Let’s look at the even money bets on a roulette table like red or black, odd or even, high or low. These bets seem like they should have a 50/50 chance of winning, just like a coin flip, right?

Well, almost. On an American roulette wheel, you have 18 red pockets, 18 black pockets, and those two pesky green ones – 0 and 00. That brings the total to 38 pockets. So your odds of hitting red (or black) are actually 18/38, or 47.37%. The casino pays you 1-to-1 on these even money bets, as if the odds were 50/50.

That tiny difference between the true odds (47.37%) and the payout odds (50%) is where the house gets its edge. Each $100 bet on red, over the long haul, will only return an average of $94.74 because of those two green pockets. The casino keeps the remaining $5.26 as profit.

Just like in the single number example, streaks can happen in the short term. You might see 10 reds in a row and think the table is “hot”. But the beauty of the law of large numbers is that as more spins are played, the results start to even out. Over thousands and thousands of spins, you’ll see red come up 47.37% of the time, not 50%.

The casino doesn’t care about any one spin or even any one gambler’s results. They’ve got math on their side. As long as they can keep the wheels spinning and the bets coming in, the law of large numbers guarantees they’ll be profitable.

In the short run, players can still beat the odds and win money through sheer luck. The casino might even lose money during a given day or week if enough players run well. But as more spins are played over weeks, months and years, the law of large numbers guarantees that the results will start to match the expected 5.26% house advantage. The casino is profitable in the long term because they have a mathematical edge, and they have the bankroll to weather any negative short-term variance.