The Law of Large Numbers (LLN) infers that the overall outcomes of a game will inevitably converge towards the true probability, if the game is played a large number of times. For example, you would expect to see almost precisely 50% heads and 50% tails, in a game of coin tossing if the game was played a large number of times.

I was recently asked for my opinion on applying the LLN to betting. The question asked whether you could accurately predict future outcomes of a game, if you discovered a moment when the prior results had not yet reached the expected probability based equilibrium, or had deviated from it temporarily. For example, if there have been more heads than tails, in a game of coin tossing, can you commence playing at this 'fortuitous' moment and expect to see more tails in the future until a 50%/50% equilibrium is reached?

I'll continue to use coin tossing in this discussion, but obviously the questioner is actually wondering whether the proposal works for traditional gambling games such as the blackjack, roulette etc. Any game will do to understand the concept, so we'll use coin tossing to keep it simple.

I am not sure that my explanation is perfectly worded, but hopefully it makes sense...

The short answer:

The short answer is no! You cannot avoid the Gamblers Fallacy by simply extending the total number of times you play a game (e.g. flip a coin). Nor will it help to extend the number of times you observe the game being played before you join in. If you extend the size of the data set, you’ll merely be extending the number of times that you must keep playing in order for the large numbers to reach equilibrium. So you run out of funds waiting for a win.

The long explanation (yawn!):

Consider that a very large data set will probably also have longer deviations from equilibrium (if not in percentage terms perhaps, but certainly in absolute numbers at least). So for example lets imagine that we have observed 10000 coin tosses, and that there has been 0.5% too many heads (50) due to random temporary fluctutations. So a data set of 10000 coin tosses cannot be expected to progress to equilibrium within 10 rolls of the dice, because you can imagine that it will take (at least) 50 tosses to reach equilibrium, actually in reality probably many many more than 50 times. Furthermore there is no guarantee that equilibrium will be reached in your given period of activity, especially when funds are limited. If your funds are limited to 10 losses, you'll be bankrupt if the subsequent outcomes deviates by only a tiny overall percentage in the wrong direction.

Now imagine the game after 1 million tosses and suppose it is now out of equilibrium by only 0.1%, it would have 1000 too many heads at that current moment! So you can't play it all the way to equilibrium because you'll either have to go to bed or run out of funds! So essentially the more times that the game is played, potentially the harder it becomes to make money!

Now let analyse it in more abstract terms. Each individual toss, or series of prior tosses that you observed, remains statistically independent of other tosses, only the data set as a whole can comply to the law of large numbers, not parts of it. Unfortunately you can’t bet on the whole data set as a single bet, you can only bet on parts of it, and you can't predict the parts of it accurately, after all they must be smaller, not large! The outcomes during the small period when you are playing a game of chance, are not influenced by the previous outcomes or the deviation from equilibrium no matter how large the prior data set - this is a key concept in probability theory. A misunderstanding of that concept leads to the gamblers fallacy.

This part of the Wikipedia entry, tries to explain this:

"It is important to remember that the LLN only applies (as the name indicates) when a large number of observations are considered. There is no principle that a small number of observations will coincide with the expected value or that a streak of one value will immediately be "balanced" by the others.”

In conclusion, I don’t believe such a gamlbing strategy based on the Law of Large Numbers will be useful. Nor is it worth trying to overcome this problem by applying a martingale staking system!

Click here for some ideas to help you find 'value' and create a profitable strategy?

Best wishes,

Gavin