3-5-7 as easy as 1-2-3

You can't get much simpler than 3-5-7. The player places 3 equal wagers, betting on the results of a 3-card poker, a 5-card poker hand and a 7-card poker hand. In each case, the Player is betting against a paytable, not anyone else's hand. In reality, the 3rd wager for the 7-card hand is optional, but as I'll show shortly, this bet has the best payback of the three. Additionally, the Player has the option to surrender half this bet after seeing the first three cards. As the math will show, this is not an option that should ever be selected.

In essence, the Player is making three independent wagers that can be analyzed independently. The first wager is identical to Pair Plus of Three Card Poker. The Player receives three cards and wins if he is dealt a Pair or Better. As there are only 22,100 unique 3-card hands, it's very easy to create a computer program which will tell us the frequency of each type of hand. Because Straights occur less frequently then Flushes for 3-card hands, we find that the payout for Straights is higher than that of a Flush.

Our program shows the following frequency of hands, and from this, we can calculate the overall payback, which is 96.5% (21,328 divided by 22,100). The player will have a winning hand just a little over 25% of the time.

The remaining four cards are community cards. The Dealer will turn over the first two of these to complete each player's 5-card hand. This wager wins if the Player has at least a Pair of 6's. Also, as Straights are now more common than Flushes, they return to their 'normal' position on the paytable. As it turns out, this wager has the worst payback of the three wagers, as it is only 95.88%. Ironically, though, the Player will win more than 36% of the time. The problem is that nearly 80% of these wins will be that of a High Pair, resulting in a win of just 1 unit. The following table shows the distribution of winning hands for this wager:

The Dealer will then turn over the final two cards. The Player is a winner if he can make a 5-Card hand that contains at least Two Pair with the High Pair being 10's or Better. This wager has a payback of 96.7%. The Player will win just over 30% of all hands. Half of all winning hands will be Two Pair, which wins just 1 unit. Three of a Kind pays 2. Straights pay 3. Flushes pay 4. Full Houses pay 5. Four of a Kind pays 7. A Straight Flush pays 20, while the Royal will pay 100. You can expect a little more than one-third of Two Pairs to be losers, not containing at least a Pair of 10's as the High Pair.

The last part of the 3-5-7 equation that most people will ask at this point is whether or not there is a time that you would want to surrender the '7' wager if the first three cards are really, really bad. I'll admit, I was sure that there would be times it would make sense. But the math doesn't lie. If you're going to surrender half of the wager, the hand would have to be so bad that you expect to win less than half your coins back, or to put in another way, it would have to have an expected value (EV) of less than 0.50. After looking at all the really bad possible hands, it turns out it gets no worse than 3 unsuited cards, all below a 10 (no Ace) that do not make a three card straight (i.e. an unsuited 2-7-9 or 2-6-9). A computer program that runs through all the possible draws to calculate the expected value finds that even these awful hands have an EV of 0.508, just barely enough to warrant not surrendering.

The bad news is that games that require no strategy tend to have lower paybacks than those that do. 3-5-7 is no exception. For a game with no strategy, its overall payback of 96.37% isn't as bad as some others. Given that the game requires a 3-unit wager, playing the game at a $5 table requires a $15/hand wager and can cost you about $17 per hour, which is a bit steeper than most. The good news is that 3-5-7 is a very simple game that provides good entertainment while requiring no strategy.