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# The house has the edge in 21 + 3

8 May 2012

Calculating odds in casino games is mostly pretty straightforward arithmetic. There are different degrees of complexity, and there certainly are traps. Goodness knows I've fallen into a few myself.

The devil is in the details, and a reader named Jason missed a fine point recently when he e-mailed me to say he thought players have an edge on the blackjack side bet 21 + 3. Your first two cards and the dealer's face-up card are used to make a three-card poker hand. If the resulting hand is a flush, straight, three-of-a-kind or straight flush, you're paid at 9-to-1 odds.

Jason wrote, "Being that they use a 6-deck shoe, if you just figure the odds of making a three-card flush, wouldn't that come out to 9-to-1? The way I figure it, the odds of two cards being the same suit would be 3-to-1. If your first card is a diamond, there are three suits that ruin your flush chances and one card that helps (hence 3-to-1 odds). So if the odds of two cards being the same suit are 3-to-1, wouldn't the odds of three cards being the same suit be 9-to-1?

"Now if you factor in straight possibilities (which do occur often) and three-of-a-kind possibilities (which are very rare), this would shift the odds to the player's favor. I think a 9-to-1 payout should be offered just if all three cards make a flush."

"I'm asking this because I've played this gimmick bet on about 10 different trips to their casino, and every single time I have profited from this bet. I know that a casino would never offer a wager which favored the player, but maybe in this case they do?"

He went on to suggest that perhaps the side bet was there to entice customers to play blackjack, since they must make a 21 bet in order to also make the + 3.

Jason arrived at 9-to-1 by taking 3-to-1 odds, then squaring them. But 3-to-1 is the same as saying 1 chance in 4, and that's the figure you must square -- leaving alone, for the moment, the problem of dependent probability. There is a 1 in 1 chance your first card will be some suit that could lead to a flush. There then is a 1 in 4 chance your next card will be of the same suit. Multiply 1 in 1 by 1 in 4, and you see there's a 1 in 4 chance the first two cards will be of matching suits.

To go the final step to a three-card flush, multiply the 1 in 4 chance of the first two cards matching by a 1 in 4 chance the next card will also be in that suit. That gives you a 1 in 16 chance at a flush or straight flush, which is the same as saying 15-to-1 odds.

The actual odds are a little longer that, because the probabilities change with each card removed from play. If your first card in a six-deck shoe is a heart, then one of the 78 hearts in the 312-card deck is no longer available. Instead of 78 in 312 -- or 1 in 4 -- the chances of the next card also being a heart are 77 in 311, and chances of a third heart are 76 in 310.

So the chances of being dealt a flush aren't exactly 1 times 1/4 times 1/4. They're 1 times 77/311 times 76/310. That boils down to 1 in 16.47, and the odds against your being dealt three cards of the same suit are 15.47-to-1, which includes straight flushes.

The bottom line for 21 + 3 is that there's a 3.24% house edge. That's not bad as side bets go, but it's not the player positive Jason suggested.

MORE ON 21 + 3: Unlike three-card poker, which has a pay table that in most versions caps at 40-to-1 on a straight flush, 21 + 3 pays the same regardless on any winning hand. There's not really a ranking of hands, except for winner vs. non-winner.

Nonetheless, some winning hands are more common than others. A game breakdown at wizardofodds.com notes that in a six-deck shoe, there are 5,013,320 possible three-card combinations. Once we separate out the straight flushes from flushes and straights, there are 292,996 possible flushes, making that by far the most common winner. There are 154,520 straights, 26,312 three of a kinds, and 10,368 straight flushes.

If we were ranking hands, straights would outrank flushes, just as they do in three-card poker. In five-card poker games, flushes outrank straights because straights occur more often -- nearly twice as often as flushes in five-card stud. Not so in three-card games. Instead, flushes come up nearly twice as often as straights.

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John Grochowski

John Grochowski is the best-selling author of The Craps Answer Book, The Slot Machine Answer Book and The Video Poker Answer Book. His weekly column is syndicated to newspapers and Web sites, and he contributes to many of the major magazines and newspapers in the gaming field, including Midwest Gaming and Travel, Slot Manager, Casino Journal, Strictly Slots and Casino Player.

Listen to John Grochowski's "Casino Answer Man" tips Tuesday through Friday at 5:18 p.m. on WLS-AM (890) in Chicago. Look for John Grochowski on Facebook and Twitter @GrochowskiJ.