Those 3-Card Straight Flushes in Let It Ride

The casino game Let It Ride by Shuffle Master Gaming, Inc. enjoys a house edge of between two and three percent against a knowledgeable player and much more than that against other players. Nevertheless, it is fun to play and seems to have earned a permanent place in the gaming world. The optimal strategy for playing the game is easily obtained and is easy to master. For example, in the May 1999 issue of Casino Player, Henry Tamburin published an article titled "The Basics of Let It Ride" in which he gave the correct strategy for the game. There is also a booklet by Stanley Ko called Mastering the Game of Let It Ride, available in most gambling book outlets and from Huntington Press, that gives the correct optimal strategy for the game. Although it is considered a no-no in the game to look at another player's hand, this rule is not always enforced and so Stanley gives some advice on how to use this additional information when it is available.

If you are a regular reader of this column, you know that my goal here is generally to address the question of "Why?" and not "How?". This article will also be in that spirit. Specifically I want to address reasons why three-card Straight Flushes on the first hand are optimally played the way the aforementioned authors recommend. First, however, let me very briefly review the game for you.

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 deck of 52 cards and, not surprisingly, the deck is shuffled with a Shuffle Master automatic shuffler. 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 finally form a five card hand. The payoffs in the game are based on standard Poker hands. 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 payoff schedule.

Notice that compared to Video Poker, 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/her fate.

The 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, 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. Once the decisions on bet number 2 have been made, the dealer turns over the second community card and the payoffs, if any, are made. Bets not receiving a payoff are losers and are collected by the dealer.

In this article I want to look at a very specific question and see how it can be answered. Suppose that before seeing any of the community cards, the player holds a three-card Straight Flush, that is, three suited cards that can be completed to form a five-card Straight Flush. Should the player pull back bet number 1 or Let It Ride and why?

Let us begin by noting that with such a hand it is impossible to obtain Four of a Kind or a Full House. Also, unless all three cards are 10 and higher (called a Three-Card Royal), a Royal Flush is impossible. Not surprisingly, one should let a Three-Card Royal ride, so we are not going to consider this type hand except as an afterthought to our analysis.

Here is our game plan. A folded hand has an expected return of 0. Thus, if letting a hand ride has a positive expected return, one should let the hand ride; if it is negative, then one should fold the hand. 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

where n! = n(n - 1)(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. Since there are only three cards known to us when the first Let It Ride decision is made, there are 49 cards remaining from which the two community cards could occur. In other words, using formula (1), there are C(49, 2) possible ways to complete our hand. This number is:

We will make use of this number below.

Let's start with a suited Straight Flush having no gaps, no high cards, and two open spots to fill at each end. An example of this would be the 3, 4, 5 of Spades. Note that there are three ways that this hand could be completed to form a five-card Straight Flush, namely A2345 or 23456 or 34567 of Spades. How manys Flushes are there?. Well, there are 10 Spades remaining, so there are C(10, 2) = 45 Flushes possible. Remember, however, that three of these are Straight Flushes and have already been counted, so there are actually 45 - 3 = 42 ordinary Flushes. It will be worth remembering that the number of Flushes for the type of hands we are considering will always be 45 minus the number of Straight Flushes. How many Straights? Well, there are three types of Straight, namely, those starting with a low card of Ace, 2 or 3. Each of these occur the same number of ways. Ace low, for example, occurs as follows. There are four Aces remaining and we want to pick one, and for each such choice there are four possible twos, so altogether there are 4 x 4 = 16 such Straights. Recall, however, that one of these is a Straight Flush so there are really only 15 ordinary Ace-low Straights. It is easy to see from this calculation that the number of Straights will always be 15 times the number of different types of Straights that can occur (equal to the number of Straight Flushes, by the way).

The Three of a Kind is calculated as follows. One of the cards in the triple has to already be in the three card hand. Hence, there are 3 choices we can make and for each choice there are C(3,2) ways to pick two of the remaining three cards of that type. It follows that the number of triples is 3 x 3 or 9. Again, for two pair, the cards forming the two pair must already be in the three-card hand. We must choose two of the three cards in the hand and for each such pair we must choose one card of each type from the remaining three of that type. In other words, there are 3 x 3 x 3 or 27 two pair hands. Finally, for the ten or better hand, we have to choose a pair of cards that are not in the three card hand. There are five packets of high cards left in the deck, and each packet contains four cards. We choose one packet and then two of the four cards in that packet. There are C(5, 1) x C(4,2) ways to do this, or 5 x 6 = 30.

Before we go any farther, let us use the above calculations to make some general observations about the number of ways a particular five card hand will occur when drawing to a three-card Straight Flush.

• The number of Flushes will always be 45 minus the number of Straight Flushes.
• The number of Straights will always be 15 times the number of Straight Flushes.
• The number of Triples will always be 9.
• The number of Two Pairs will always be 27.

The total number of winning hands in the example at hand is 3 + 42 + 45 + 9 + 27 + 30 = 156, so the number of losers is 1176 - 156 or 1020. We can now calculate the conditional expected return on this hand as follows:

Note that in Figure 2 I left the 1176 denominator out of the probability column since we can divide by it after all the other calculations have been made. The expected return by letting the suited 345 ride is 252/1176 or approximately 0.2143. This means that for a dollar wager, this hand is worth, on average, about 21.43 cents. We should let the hand ride.

What about a suited 234? Here the number of Straight Flushes is only 2, namely A2345 and 23456. The number of High Pairs will be 30 as it was above. Making use of our observations above we can easily fill in the conditional return table.

Notice that now our expected return is -30/1176 or about -2.55 cents; the hand should be folded. The suited A23 seems even worse in the sense that there is only one Straight Flush. On the other hand, the Ace in the hand raises the number of High Pairs as follows. There are four four-card high packets left in the deck. If we choose one of these, then there are C(4, 2) or 6 ways to choose two of these cards. Altogether, then, there are 4 x 6 or 24 non-Ace High Pairs. For the pair of Aces we can choose one of the remaining three Aces and then any one of the 40 cards left that are neither Aces, Twos, or Threes. In other words, there are 3 x 40 or 120 Ace pairs. Altogether, then, there are 144 High Pairs. We obtain the following table.

The increase in the High Pairs here is not enough to offset the single Straight Flush and only fifteen Straights. The hand is worth -7.14 cents and should be folded.

Well now, what about Inside Straights. Clearly if the low card in such is an Ace or a Two, the hand is going to be worse than the hands we just looked at; these should be folded. What about a suited 457? Here there are two Straight Flushes, namely 34567 and 45678. As in the first calculation we did for the suited 234, there are 30 High Pairs. Again using our previous observations, the table for this hand is exactly the same as it was for the suited 234; the hand should be folded. What about an Inside Straight with a high card? Let's look at a suited 89J. Here there are 2 Straight Flushes but, as was the case with the A23 hand, there are 144 High Pairs. The table looks like this.

This hand is worth 198/1176 or about 16.84 cents; you should let it ride. Obviously, if an inside Straight Flush has more than one high card you should Let It Ride.

Double Inside Straight Flushes? I'll leave it to you to show that if such a hand has only one high card, then it should be folded (look to my A23 calculations for help). What if it has two high cards? Let's look at 9JK. Here there is only one Straight Flush. There are 3 x C(4, 2) or 18 High hands consisting of 10, Queen, or Ace. As in our calculation for the A23 or 89J, there are 120 Jack pairs and 120 King pairs. Altogether, then, there are 18 + 120 + 120 = 258. Here is the table.

Surprise, this is not a bad hand. It is worth 144/1176 or about 12.24 cents on average. You should let this hand ride.

If you think about what we have done, these calculations settle all of the three-card Straight Flush hands. Clearly a Double Inside Straight Flush containing three high cards is better than one containing only two high cards, so it follows that you should let any three-card Royal ride. By doing calculations similar to the ones above, you can easily convince yourself that hands lower than a three-card Straight Flush don't cut it. If you have questions about this, let me know and I'll come back to it in a later column. Obviously you should let any winning three-card hand ride.

So there you have it: the correct strategy for three-card Straight Flushes in Let It Ride. All nice and neat and there wasn't a computer in sight! See you next month.

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