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# Let It Ride (Again)

6 May 2001

Last April I published a column called Those 3-Card Straight Flushes in Let It Ride on this web site. In it I showed the reader how to do the calculations that determine the correct strategy for playing straight flush hands on the first (three-card) hand of the game. In this article I would like to review those results, determine the rest of the correct plays for three-card hands, and then show you how to derive the correct playing strategy for the four-card hands. First, however, I think it would be helpful to review the rules of the game.

Let It Ride is played on a table similar to a standard blackjack table and holds up to seven players. It is played with a standard 52-card deck and is shuffled with a Shuffle Master automatic shuffler (no surprise since Shuffle Master owns the game). In front of each player are three betting circles labeled, from right to left, 1, 2, and \$. Before the deal, each player places a one-unit bet on each of these three circles. For purposes of discussion, I will assume that the unit is \$1, although it is doubtful that you will ever find such a small stakes game.

After all bets are down, the dealer deals three cards face down to each player and also two cards face down in two rectangles placed in front of the chip tray; these latter cards are called community cards and will be used together with the player's three cards to form a final five-card poker hand. Notice that the dealer does not have a hand and that is simply because the players are not competing against the dealer. Rather, they are trying to get final five-card poker hands that are paid, as in video poker, according to a payoff schedule. In Figure 1 below I give the Let It Ride payoff schedule.

 Final Hand Payoff Royal Flush 1000 to 1 Straight Flush 200 to 1 Four of a Kind 50 to 1 Full House 11 to 1 Flush 8 to 1 Straight 5 to 1 Three of a Kind 3 to 1 Two Pair 2 to 1 Pair of 10s or Better 1 to 1
Figure 1
Let It Ride Payoffs

Notice that compared to most video poker games, this seems like a rather generous pay table. This is deceptive because in video poker one gets to improve one's hand by discarding and drawing new cards; not so in Let It Ride. As I'll explain now, however, the player does have some control over his or her fate.

Action begins with each player looking at his/her three-card hand; the community cards are hidden at this point. On the basis of this information, the player can let the bet in circle 1 ride by placing the three cards face down on the table or can pull the bet back by scratching the three cards on the table (similar to calling for a hit in blackjack). Once the decision on whether or not to 'Let It Ride' on circle 1 is made, the dealer turns over one of the two community cards. Now each player gets to decide whether or not to let bet number 2 ride. Note that even if bet number 1 is riding, bet number 2 can be pulled down and vice versa. Once the decision on bet number 2 has been made, the dealer turns over the second community card and the payoffs, if any, are made. Bets riding and not receiving a payoff are losers and are collected by the dealer.

Notice that on hands 1 and 2, if one pulls back a bet the expected return on the hand is zero. Thus any hand having a positive expected return should be played and any hand having a negative expected return should not be played. For a discussion of expected return see my July 1999 article, Right Question, Right Answer (Hopefully).

Recall that the number of ways to pick k objects from a set of n objects is the number

C(n, k) = n!/[k!(n - k)!]

where n! = n x (n - 1) x (n - 2) ... 3 x 2 x 1 (and 0! = 1). For a detailed discussion of this formula, see my August 1999 article Oh New York, Bring Back Those Big Dippers.

My April 2000 article essentially established the following facts. There are C(52, 3) or 22,100 three card hands that one can deal from a 52-card deck. For each such three-card hand there are C(49,2) or 1176 ways to complete the hand. Altogether, therefore, there are 22,100 x 1176 or 25,989,600 different situations that can occur. This number will be important to us later on in next month's article.

Not all 25,989,600 situations need to be checked to determine the optimum three-card strategy; in fact, most do not. My April 2000 article reported the following expected returns for all 1176 completions of each hand:

 Type of Hand Expected Return Suited 345 +252 Suited 234 -30 Suited A23 -84 Suited 457 -30 Suited 89J +198 Suited 78J -84 Suited 9JK +144
Figure 2
Some Three Cards to a Straight Flush

Notice that with this table it is clear that any outside straight flush (3 ways to fill) whose lowest card is, necessarily, 3 or larger should be played. Also the table shows that a single inside straight flush (2 ways to fill) having one or more high cards should be played; with no high cards the hand should be folded. Finally, any double inside straight flush (one way to fill) having at least two high cards should be played; with zero or one high cards the hand should be folded. A corollary to all of this is that any three cards to a royal flush should be played.

Since it is obvious that any made hand should be played (a high pair or three of a kind) that leaves ordinary 3 card flushes, three card straights, and low pairs to consider. As a reminder of how the calculations are made, let me show you the table one constructs to calculate the return for a low pair.

 Hand Number Payoff Product Royal Flush 0 10000 0 Str. Flush 0 200 0 Quads 1 50 50 Full House 9 11 99 Flush 0 8 0 Straight 0 5 0 3 of a Kind 88 3 264 2 Pair 198 2 396 High Pair 0 1 0 Losers 880 -1 -880 Totals 1176 -71
Figure 3
Expected Return for Playing Low Pair in Hand 1

Clearly we should fold a low pair in Hand 1. Playing an unsuited TJQ, a straight with 3 high cards, produces an expected return of -27. An ordinary flush with 2 high cards such as 6TJ suited yields an expected return of - 138. From these results it is clear that all low pairs, all flushes, and all straights should be folded at Hand 1. We therefore have a complete optimal strategy for playing Hand 1:

• Play any 3 cards to a Royal
• Play any straight flush with low card greater than or equal to 3
• Play any single inside straight flush having at least one high card
• Play any double inside straight flush having 2 high cards
• Fold anything else

Here is a caveat. Suited low straight flushes containing an ace are double inside; suited 234 is a single inside straight flush. Both should be folded.

Now what about Hand 2? There are C(52,4) or 270,725 four-card hands one can obtain and each of them can be completed in 48 ways. Here for example is a table that shows how to compute the expected return for an inside 4-card royal.

 Hand Number Payoff Product Royal Flush 1 1000 1 Str. Flush 0 200 0 Quads 0 50 0 Full House 0 11 0 Flush 8 8 64 Straight 3 5 15 3 of a Kind 0 3 0 2 Pair 0 2 0 High Pair 12 1 12 Losers 24 -1 -24 Totals 48 1067
Figure 4
Expected Return by Playing Inside Royal in Hand 2

Here, tabulated, are the results of similar calculations. As you'll see, there is enough information in the table to determine the correct play/fold strategy for Hand 2.

 Type of Hand Expected Return Inside Royal +1067 Str. Flush In. 0 High +243 Flush with 0 High +33 Outside Straight 0 High 0 Outside Straight 1 High +6 Inside Straight 4 High 0 Inside Straight 3 High -6 Low Pair -22
Figure 5
Various Expected Returns for Hand 2

Notice that since a flush with 0 high cards plays, we really didn't need the first two entries. Since any made hand obviously plays we have the following optimum strategy rules for Hand 2:

• Play any four cards to a Royal Flush
• Play any four cards to a Straight Flush
• Play any four cards to a Flush
• Play any Straight with one high card
• Play any Straight with no high cards (optional)
• Play any Inside Straight with 4 high cards (optional)
• Fold everything else

The sixth and seventh items above explain why optimal strategy on Hand 2 varies from writer to writer. Both of the optional plays are plays with 0 expectation so the expected return from playing is exactly the same as the expected return by standing. Because playing these hands produces a higher average bet, the expected return per unit wagered is slightly lower if these are played than if they are not. This is analogous to the odds bet in craps. The edge in craps is lowered when taking odds not because the loss per game is lowered but because the average bet per game is raised. The rationale is that if you're willing to play a negative game, then you ought to be willing to take a fair bet. It's really up to you, but I would take these bets just to get more action into the game.

Notice how little we had to calculate to derive an optimal playing strategy for this game. Usually calculating an optimal strategy for a game is the hard part and figuring the house edge from this is the easy (or easier) part. Let It Ride is the other way around. Next month I'll show you how to calculate the house edge in this game and you'll see that there are lots of numbers to deal with.

If you have questions about this or any other of my articles, or if you just have questions about other casino games, you can reach me by clicking on the word Technigame in the 'About the Author' box at the end of the article. This will link you directly to my email and you can reach me that way. See you next month.

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Donald Catlin

Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

#### Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers
Donald Catlin
Don Catlin is a retired professor of mathematics and statistics from the University of Massachusetts. His original research area was in Stochastic Estimation applied to submarine navigation problems but has spent the last several years doing gaming analysis for gaming developers and writing about gaming. He is the author of The Lottery Book, The Truth Behind the Numbers published by Bonus books.

#### Books by Donald Catlin:

Lottery Book: The Truth Behind the Numbers