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I'm writing a paper on the game of blackjack and I'm trying to make a point about how a common player's attitude is a logical/statistical fallacy in the following situation:

When any common person steps up to a blackjack table, they realize that the odds are slightly in the dealer's favor (they're correct), but they also realize (correctly) that there's enough variability from the mean on any given hand for them to make some profit. Now they still won't bet a massive amount on any given hand, fearing that that variability could swing in the negative direction rather than the positive, but they might settle upon a smaller, "safer" bet that they'll make consistently. Their assumption (and this is where the fallacy comes in) is that they'll be able to make a large profit over time by betting a small amount on each hand and taking advantage of the large variability on each given hand.

In reality, as they play more and more hands, that standard deviation which overshadowed the mean cash gained for a single hand begins to dwindle in comparison to the mean. This is because the standard deviation increases proportionally to the square root of the number of trials whereas the mean increases proportionally to the number of trials. So, in the long run, the player will no longer be able to take advantage of that high volatility and will be in fact doomed (with a certainty of essentially 1) to lose money.

I realize this is a rather technical scenario (maybe more statistics than philosophy), but might there be any formal name for this sort of fallacy? (Or maybe a snappy-sounding name I could construct?)

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It's just the fallacy of composition. The person is reasoning: It is possible for me to make money on a single hand, therefore it is possible for me to make money on a (sufficiently large) set of hands. The inference is incorrect, for the reason you mention above.

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