I was teaching beginner 9/6 Bonus Poker Deluxe in my Tuesday noon class at the South Point. (The semester will continue every Tuesday until September 4.) The hand we were discussing was 3♦4♦5♦Q♣J♣.
The class is taught using a top-down strategy where you select the rule that comes first. Holding 3♦4♦5♦ (along with certain other 3-card straight flushes) was covered by Rule 8. Holding Q♣J♣ (along with the other two suited high card combinations) was covered by Rule 9. Since Rule 8 comes before Rule 9, the correct play is to hold the 3♦4♦5♦.
One student asked the question: “But wouldn’t I at least get my money back more often if I held the clubs?”
My answer was: “Absolutely. But the system we use to determine the correct play maximizes Expected Value. With Expected Value, it is the frequency of the win multiplied by the value of the win – not just the frequency.”
The following chart shows this. To make the numbers big enough to read easily, I had to split the chart in half.
The Expected Value of 3♦4♦5♦ is shown to be 3.025. Since you are drawing two cards, there are 1,081 different combinations you could draw. About 87% of the time (actually 937 out of 1,081) you end up with no win at all.
But of the times you do score, most of them are straights and flushes, paying four and six times the value of high pairs respectively.
All the numbers in the preceding paragraph came from either the pay schedule or the chart below — which was copied directly from the Video Poker for Winners software. If you wish to be able to understand simple video poker mathematics, this is a good chance for you to practice.
| Holding | EV | Total | No Win | High Pair | 2 Pair | 3K | ST |
| 3♦4♦5♦ | 3.025 | 1,081 | 937 | 18 | 27 | 9 | 45 |
| Q♣J♣ | 2.9374 | 16,215 | 9,827 | 5,022 | 711 | 281 | 189 |
| Holding | EV | Total | FL | FH | 4K | SF | RF |
| 3♦4♦5♦ | 3.025 | 1,081 | 42 | 0 | 0 | 3 | 0 |
| Q♣J♣ | 2.9374 | 16,215 | 162 | 18 | 2 | 2 | 1 |
When you start from Q♣J♣ and draw three cards, there are 16,215 possible draws. This number is exactly 15 times as large as the 1,081 possible draws when you only draw two cards.
You get a high pair or two pair 5,733 times out of the 16,215 (which is about 35% of the time), but these are only 5-coin wins. You score something about 40% of the time, but most of the wins are small.
Other players use the logic that holding clubs gives them a chance at a royal flush and holding the diamonds doesn’t. But a 1-in-16,215 chance at 4,000 coins is only worth about 0.24 coins. The 3-in-1,081 chance of getting a straight flush holding 3♦4♦5♦ is worth 0.69 coins and that is something usually omitted by seat-of-the-pants players thinking, “It seems to me.”
A lot of players try to reason correct plays out in their heads. While this is certainly an appropriate avenue to address the problem if you don’t have a strategy handy, correct strategies are fairly easy to come by and figuring out how many times in 1,081 or 16,215 (or even bigger numbers when you draw four or five cards) is a tedious, error-prone process and basically impossible for most people to do by themselves.
A computer program, however, can figure this out almost instantaneously and very accurately. It’s one of the tools of the trade that makes it possible to play well.