Let it Ride Probabilities

Let it Ride is really a game of three bets and each is independent of the other so the calculation of the house edge is really a calculation of each betting phase and then summing them up.

Whether or not a player uses the above strategy, the winning hands will be dealt with a certain frequency. The tables below show that frequency for three- four- and five-card poker hands. Each table gives the number of card combinations that comprise a specific hand, and its probability of occurring. The next column shows the total payoffs that can be expected for how this hand can develop in the next two cards.

For example, if the three dealt cards are three of a kind, then there are some things we know. After three cards, it’s a pat trip. After four it could be four of a kind, and if not, it still could improve on the last card. Moreover, the last two cards could be a pair, creating a full house for this hand. So the “total payoffs” are a sum. It is the total of the combinations creating a full house (72) times the payoff for that (11), plus the combinations creating four of a kind (48) times the payoff for that (50) plus the total combinations creating three of a kind (1176 minus 72 & 48) times the payoff for that (3).

It is important to avoid double counting, so even if all 1176 combinations result in three of a kind (because it’s already there), that number should be reduced by the number of combinations used in adding up the four of a kind and full house possibilities. This is tedious and tricky. The right answer is 6,360 bets. This is the total possible “productivity” of this hand in all its combinations of 1176 two-card deals.

For another example, take a beginning hand, like three cards to a straight flush with no gaps and no high cards. Qualifying improvements would include a high pair (off the board), two pair (having each board card match a card in the hand), or three of a kind (a pair off the board that matches one card in the hand). These calculations result in pretty low total payouts at this level, but they increase in relative importance as cards are turned over and they creep closer to having real value as straights and flushes.

The average payout per combination is the total payout number divided by the number of possible combinations, which is 1176. This says that “on average” the next two cards should generate a certain return. In the case of three of a kind, it is 5.4082 bets. This number, times the probability of getting three of a kind in the first place, yields the expected profit that three-of-a-kind can generate for a player at the three card level. This number is 0.01273 bets.

The sum of all these “profit” expectations gives the total expectation from the three-card-level of play for those hands that generate a raise, or if you prefer, a “Let it Ride.” This number is 0.119145 bets. Now a raise has been made in only 7.28% of the hands, so the “average” bet under consideration is 0.0728 of a bet, and it generates a return of 0.119145. This means that the player’s expectation is positive at this juncture.

It makes sense that the expectation is positive, as the player has the option to flee the additional bet if the hand will not support it. So in the 92.7% of hands that do not show any promise, the bet has been withdrawn.

A similar table is presented below for the four card level. It has a couple of rows that the three card table did not have because two pair, straights and flushes are all now viable hands. The total payout and average payout columns have been calculated the same way.

The bottom of this table shows that the average expected return at the four-card level for persons with hands qualifying for the extra bet is 0.29886 betting units on an average bet of 0.15932. Again, the expectation is positive, as the players have to option not to raise in this round with cards that do not support it.

Finally, there is the five-card table, in which all cards are known, and the hands have no way to improve further. The bottom line at the end is negative. The average bet is 1.0, as it is mandatory, and the average profit expectation is -0.37272 betting units. It is obvious that the high percentage of losing hands turns the situation away from the player and towards the house.