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For example if I bet $5 and get a blackjack that pays 3:2, this means that I should get payed 7.5.

The problem is that the there is no $0.5 dollar chip. How this situation is handled in the casinos?

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From Wikipedia:

$2.50 chips are mostly used for blackjacktables, since a "natural" (a 21 on the first two cards dealt to a player) typically pays 3:2 and most wagers are in increments of $5. However, the Tropicana Casino and Borgatain Atlantic City, New Jersey, and others, have used $2.50 (pink) chips in $7.50 to $15 and $10 to $20 poker games.

They have $2.50 chips especially for the case you're describing.

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I don't know how American casinos handle this, but in Austria there is the same problem with euros. There is no 0,5 euro chip, so you simply get a regular 50 cent coin.

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It really depends. I've seen some casinos pay 7.50 with a 5, two 1s and a half-dollar coin, I've even seen casinos that actually had a $2.50 chip. These were all in California.

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I've been to casinos in Las Vegas and Wisconsin and I think they have all had 50-cent chips at the blackjack table. If they don't, you can work around the problem by betting an even amount. For example, playing 6 dollars a hand at a table with a 5-dollar minimum won't have a significant impact on how much you win or lose.

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In Las Vegas, the craps table was very clear that they round down to the nearest whole dollar. I can't tell you if that's based on casino policy and/or local law, and you didn't state where you are.

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