The math of match play

All casino games are math problems at heart, even if you can't see the arithmetic going on or use it to your advantage.

Slot machine industry professionals, including manufacturers and the casino slot directors who decide which games you see on casino floors, talk about games having the right math model. The frequency of winning spins, the size of wins, frequency of bonus rounds to keep players interested while generating a profit for the house — that's in the math.

For players, basic strategy at blackjack is rooted in math. So is counting cards, of course. Calculation involving frequency of cards drawn is behind strategy in video poker.

Math is behind the house edge at every game, and factors into the comps, perks and giveaways casinos use to keep you coming back. Something as simple as a match play coupon all comes down to the numbers — the casino is calculating that giving you a mathematical edge on one wager will entice you to stay and play more hands with the house's usual mathematical edge.

How large an edge do you get on that one match play wager? That's a topic that came up recently on the Boyd's Eye View, video poker author Linda Boyd's forum at the Midwest Gaming and Travel magazine site, midwestgamingandtravel.com. I check out the forum every now and then, and found a question about using match play.

Most of you have seen match play coupons. When you use one and win, you get double winnings for that one play. Bet $5 and a match play on an even-money bet, and if you win, you get $10 in winnings, you keep your $5 bet, and the dealer takes away the coupon. Lose, and you lose $5 and the dealer also takes the coupon.

It's a pretty sweet deal, and a poster on the forum wanted to know just how sweet.

"If I make a pass line bet at craps with no added odds the house advantage is 1.41%. What happens if I make that bet using a match play?

"If the house had no advantage at all on the pass line, the match play would give me a 50% advantage. So is it 50% minus that 1.41%, for 48.59%? Or am I computing too simplistically?

It's not quite that simple. I replied, and took the long way around in the calculation, just making sure everyone understands why the calculation is valid.

The 1.41% house edge on the pass line in craps means we win 49.295% of the time, and lose 50.705%. Subtract 49.295 from 50.705, and you get 1.41.

So let's say you wager $5 on the pass line for each of 100,000 comeout rolls. You risk $500,000, and your expectation is that you'll win 49,295 bets and lose 50,705.

Each time you win using the match play coupon, you get $10 in winnings, plus you keep your $5 bet. So after 100,000 trials, you have 49,295 x $15, or $739,425. Subtract the $500,000 that represent your wagers, and that means you have a profit of $239,425.

Divide that by $500,000 in wagers, then multiply by 100 to convert to percent, and your edge over the house is 47.885%.

If you had an unlimited supply of match play coupons and never made a bet without one, you would have a 47.885% edge over the house. You'd expect to go home with a profit very close to 47.885% of the sum of your wagers.

Casino operators know the math — better than most of their customers — so they won't let you play coupon after coupon after coupon. But for the moment your match play is on the table, it sure is nice to have the math on your side.

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Speaking of casino math, a video poker player e-mailed to ask about a situation in 9/6 Jacks or Better video poker.

Dealt a hand such as ace of hearts, king of clubs, jack of spades, 7 of diamonds, 3 of spades, expert strategy says we should hold the King and Jack while discarding the other three. "Why not ace-king?" the reader asked.

The key is in the frequency of straights. With any three-card draw, there are 16,125 possibilities. Holding either ace-king or king-jack leave us with 4,914 draws that would give us a paying pair of jacks or higher, 711 two-pair draws, 281 three of a kinds, 18 full houses, and two four of a kinds.

However, holding ace-king means the only possible straights we could draw are ace high, while holding king-jack leaves open the possibility of either ace-high or king-high straights. Holding ace-king, the possible draws include 48 straights, while holding king-jack leaves 112 possible straights. The greater straight possibilities make the strategy difference.

The strategy does change in some games that offer big bonuses on four-ace hands. In Double Double Bonus Poker, where four aces plus a 2, 3 or 4 as a fifth card pays 2,000 coins for a five-coin wager, the best play in our sample hand is just to hold the ace and draw four fresh cards. But in Jacks or Better, Bonus Poker, even Double Bonus Poker, the straights dictate our decision to hold king-jack.