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In blackjack, the probability of winning a hand played by the books is 48% (or so I've heard). Therefore the chance of winning three consecutively is 11% roughly. If I start from scratch 8 times and try to win 3 in a row each of those 8 times, what is the probability it will occur once in those 8 attempts?

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If you assume that all the hands are independent (valid since casinos use ~6 decks of cards) and you play all 3 hands at once (otherwise you might stop if you lose the first or second hand). Let $X$ denote the number of times you win 3 hands consecutively from your 8 attempts then:

$$ P(X = x) = {8 \choose x} 0.11^x (1 - 0.11)^{8-x} $$

Since $X \sim Bin(8, 0.11)$. In answer to your question $P(X = 1) = 0.3892357$. If you stop playing early on each of the 8 attempts when you lose on the 1st or 2nd hand this becomes a little different so let us know if that is what you intended.

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