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Of course I may lose my bet, but according to the law of average, I will not. [Source: A Message to Garcia]

What does " the law of average” mean? What is the meaning of the sentence? Some dictionaries include "the law of averages".

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At least nowadays, the “law of averages” is the commonly used phrase in my experience. According to Wikipedia, that work was published in 1899. “Average” might have been more common then.

In any case, they refer to the same thing:

The law of averages is the fallacious belief that a particular outcome or event is inevitable or certain simply because it is statistically possible. Depending on context or application it can be considered a valid common-sense observation or a misunderstanding of probability. This notion can lead to the gambler's fallacy when one becomes convinced that a particular outcome must come soon simply because it has not occurred recently (e.g. believing that because three consecutive coin flips yielded heads, the next coin flip must be virtually guaranteed to be tails).
(Wikipedia)

Often, people mistakenly believe that they are due to win (a certain gamble for example) because they have lost consecutively many times. They reference the so-called law above to justify their thinking.

That is what the speakers doing. Presumably he lost his bet before. Then by the law of averages, he believes he will not lose his bet again.

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