Good Problems

5 May 2006

By Elliot Frome

Your eyes light up as the cards are dealt. Ten of Diamonds. King of Diamonds. Jack of Diamonds. Queen of Diamonds. YES! Your heart sinks a little as you see the last card. Four of Diamonds. Of course, you wanted the Ace of Diamonds, but there is a part of you that would've preferred the 8 of Clubs instead of the 4 of Diamonds. With the 8, your decision would be easy. With the 4, you're not so sure what to do. Of course, it's hardly the worst dilemma in the world, given both hands have very high expected values.

You obviously have two choices. You can keep the Flush and collect your payout. Assuming you're playing max-coin dollars, you'll collect $30. Or, you can throw the 4 away, and go for the Royal. You'll have a 1 in 47 chance to win the $4000. Of course, you'll also have 6 ways to get back that Flush, 1 way to get a Straight Flush, 7 ways to get a Straight and 9 ways to get a High Pair. That leaves 23 ways to wind up with nothing. What's the right way to play it? Expert Strategy always dictates that you play the hand that has the highest expected value. When we add up the possible payouts by going for the Royal and divide by 47, we find that the expected value of 19.64. This compares to an expected value of 6.0 for the Flush. The proper play is quite clear.

A lot of people find this particular play to be very difficult. For many it is very hard to throw away guaranteed money. This should not be surprising. At the same time, if you walked into the casino and was offered the opportunity to wager $30 with an expectation that in the long run, this money will be more than tripled, most people would take it. After all, many people will wager $30 on any number of games (blackjack, Three Card Poker, a $5 video poker machine) where the long term expectation is far less. Of course, the situations are not identical. In this situation, you will basically get only ONE chance to wager your $30, as a dealt hand of a Flush that contains a 4-Card Royal is hardly a common one. In the end, many people choose to take the guaranteed money, even though it is not the proper play.

Then there are those who will point to the expected value of 19.64 and say that is only the 'long run' and the 'long run' is millions of hands. However, there are only 47 possible outcomes for this particular hand, which means it won't take millions of hands of this particular type to reach the long term. Many of these people have dismissed all the math and computer analyses and theories of long term play in favor of short-term play. Since these people have dismissed math as the basis for their reasoning, I'll attempt to use logic as a means of showing why the right play is to go for the Royal.

While each individual only cares about his own experiences, to the casino all that matters is the results of all the hands played in their casino over time. They don't care if 1 person plays a billion hands or if a billion people play 1 hand. To the casino, it's still 1 billion hands. So, the question would be, would the casino do better if every player who gets a hand like the one described above keeps the Flush or goes for the Royal. I think that everyone would agree that over a year, any sizeable casino will be happy to see all the players dealt this hand, keep the Flush.

Over a year, a casino with 150 video poker machines, being played an average of 600 hands per hour for 10 hours per day will have a total of 328 MILLION hands played. Clearly, by statistical standards this is the long term. As a ballpark figure, the 4-Card Royal within a 5-Card Flush will occur about 1 in 20,000 hands, or over this period of time, 16,425 times. Over this many hands, you can be certain that we will achieve within a fraction the theoretical expected value (all hands of this type have a combined expected value of 18.66). The casino will reap HUGE rewards if every Player chooses to keep the Flush. They'll pay out just under 100,000 coins per coins wagered, as opposed to 306,490 if all the Players choose to go for the Royal. At dollar max-coin, this translates to 1 MILLION DOLLARS less paid out when the Players keep the Flush.

Now, if the casinos pay out 1 million less, that means the Players are being paid 1 million dollars less. If the Players on the whole are being paid 1 million dollars less over a year because they all chose to hold the Flush, it would seem to follow that it makes no sense to believe that any one Player will benefit from doing so even if he is playing just one hand. He MIGHT benefit from doing it, if the resulting draw results in a return of less money, but the odds are in his favor that he will benefit by drawing.

Is there ever a time you should keep the Flush? I never say never. If you're in a tournament, there might be some reason to do it. If you're going to miss your flight home if you have to wait for a hand-pay Royal might be another. Another reason might be if you need the $30 you'll win on the Flush to buy food, but then again, you probably shouldn't have been gambling with your food money anyway.