Expectations Galore: A Reader's Question

George Taylor had written to Poker Digest complaining about the difficulty he and his wife have had in getting information on the house edge for Let-It-Ride and Caribbean Stud. In this article I'm going to determine the house edge for Let-It-Ride. It's important to emphasize the calculation is for the house edge when a player is following the optimum strategy which was given in my previous article. An important basis of the optimum strategy is the assumption the player has no knowledge about cards other players have in their hands.

Let the first bet ride in the following situations and no others:

1.

Your first three cards already win.

2.

Your first three cards are suited and allow a royal flush.

3.

Your first three cards are suited and allow three straight flushes.

4.

Your first three cards are suited, allow two possible straight flushes, and at least one card is of rank 10 or higher.

5.

Your first three cards are suited, allow one possible straight flush, and two cards are of rank 10 or higher.

Let the second bet ride in the following situations and no others.

1.

Your first four cards already win.

2.

Your first four cards are suited.

3.

Your first four cards allow an open ended straight draw and at least one card is of rank 10 or higher.

There are only two decisions to be made in Let-It-Ride: Does the player let the first and/or second bets ride. What the above strategy captures is letting a bet ride precisely when there is a positive expectation in doing so.

It is not difficult to precisely calculate the expectation of the above strategy. Since there are many subcases involved in carrying out the calculation, I will not do so here. Instead, let me provide the framework for the calculation. Full details will be posted on my website (http://www.math.sfu.ca/~alspach) in the future.

The game really has three separate bets, and these bets are not coupled in any way. That is, your decisions for the first and second bets are completely independent of each other. Thus, the expectation for the strategy is the sum of the expectations for each of the three bets.

The expectation of the third bet is the simplest to calculate. Simply multiply the probability of each type of winning hand by the return for that win, sum these values and subtract the probability of losing.

The total number of five-card hands is C(52,5) = 2,598,960. There are four royal flushes, 36 straight flushes, 624 four-of-a-kinds, 3,744 full houses, 5,108 flushes, 10,200 straights, 54,912 three-of-a-kinds, and 123,552 two pair. To determine the number of winning big pair hands, there are five choices for the rank, six choices for the pair, C(12,3) = 220 choices for the ranks of the remaining three ranks, and four choices for each card of the latter ranks. Multiplying all of this out gives 422,400 big pairs. This means there are 1,978,380 losing hands.

The probability of getting a particular winning hand is obtained by taking the number of them and dividing by 2,598,960. Multiplying these probabilities by the respective returns (1,000 for a royal flush, 200 for a straight flush, etc.) produces -.3727 which is the expectation of the third bet.

The contributions for the first and second bet are more complicated to determine. There are two steps involved in determining the contribution of the first bet to the expectation of the strategy. First we must determine the probability of letting the first bet ride, and second we must determine the average return upon letting the first bet ride. The product of these two numbers is then the contribution towards the expectation of the strategy.

Altogether the player has C(52,3) = 22,100 possible three-card hands. Of these, 52 are three-of-a-kind and 1,440 have a big pair. Thus, there are 1,492 hands which already are a winner. There are 40 hands under option No. 2 of the strategy, 28 under option No. 3, 24 under option No. 4, and 16 under option No. 5. This means there are 1,600 hands for which the player lets bet No. 1 ride. So the probability of letting the first bet ride is .0724 or about 1 in 14 times.

Next we must determine our average gain upon letting the first bet ride. The dealer has C(49,2) = 1,176 possible two-card hands. Multiplying 22,100 by 1,176, which is 1,881,600, is the total number of possible outcomes when the player lets the first bet ride. We determine the return for all of these outcomes. For example, if the player is dealt three-of-a-kind, it can become either four-of-a-kind, a full house, or not improve. It can become four-of-a-kind in 48 ways; it can become a full house in 72 ways; or it can not improve in 1,056 ways (note the possibilities must sum to 1,176 which gives a good check on whether or not one is doing the arithmetic correctly). You then multiply the number of outcomes by the appropriate return.

One does a similar calculation for each of the possible beginning hands. Some of the beginning hands allow many outcomes, one of which is losing under all but option 1, but it is easy to check correctness at each step. Adding all the numbers together and dividing by 1,881,600 gives the average return which turns out to be 1.50669. Multiplying the latter number by .0724 yields .1091 as the contribution of letting the first bet ride to the expectation of the optimum strategy

There are only three options in the optimum strategy for letting the second bet ride. However, there turn out to be many more subcases to consider under the option of letting the second bet ride if the player has four suited cards. The reason for the subcases is because with four suited cards there may be zero, one or two completions to a straight flush, there may or may not be a completion to a royal flush, and there can be a variety of big cards in the hand. Still, the principles are the same as above. Here are the relevant numbers for anyone wanting to check the calculations.

There are C(52,4) = 270,725 possible four-card hands. There are 40,865 hands for which the player lets the second bet ride. Thus, the probability of letting the second bet ride is .15095. The total number of outcomes upon letting the second bet ride is 1,961,520. The combined returns is 2,970,028 giving an average return of 1.51415 for letting a unit second bet ride. The contribution toward the expectation of the optimum strategy is then .2286.

Adding the contributions from letting the first and second bets ride to the expectation of the third bet determined above, we find the expectation of the optimum strategy to be -.035. So the house advantage is 3.5 percent of a unit per hand played. I have seen this figure mentioned elsewhere, but as I recall it was stated the figure arose from simulations. Let me emphasize the preceding calculations are exact and involve no simulation.

Let me add a few comments. It is possible to decrease the house edge but this involves knowing cards in other players' hands and adjusting your play accordingly. It is even easier (and foolish) to increase the house edge by deviating from the optimum strategy. An extremely foolish play is to let the first bet ride when holding a small pair. Another more subtle mistake is to use a betting unit which, should you make a royal flush, takes you beyond some arbitrary table limit imposed by the casino. At this point in time, I have not done any quantitative work related to the comments in this concluding paragraph.