This appendix presents my analysis of Chase the Royal video poker. The rules for Chase the Royal are as follows:
The game is based on conventional video poker. The player may choose among four different video poker variations and either 3-play, 5-play, or 10-play.
If the player is dealt a pair of jacks, queens, or kings and made a max coin bet then the player may exchange the high pair for 3 cards to a royal flush. This Chase the Royal option does not apply if the pair is part of a higher hand (for example a two pair) or if the player also has 4 cards to a straight or royal flush.
The 3 cards to a royal shall be chosen at random from among the 40 possible (4 suits by 10 ways to choose 3 ranks out of 5). The player may see the specific three cards before deciding to switch.
If the player opts to chase the royal then the other two cards will be dealt immediatly from the remaining 49 cards in the deck.
As an incentive to chase the royal a straight shall pay 10 and a flush shall pay 12 in chase the royal mode.
In the unlikely event the player receives a royal flush with maximum coins bet on all hands then each shall pay 2000, as opposed to the usual 800 per coin bet based.
As in conventional video poker the player should seek out the best pay 168bet table possible and play proper strategy. If the player has the opportunity to chase the royal the player should always do so.
Overall the player will get 4.39 as many royal flushes as conventional video poker, at the expense of much few low paying hands. The standard deviation is very high at 9.15 (for 9/6/5 double bonus), compared to the normal 5.55 for the same game. This game is well suited for the player who puts an emphasis on excitement over bankroll preservation.
9/6/5 Double Bonus
So far the best pay table I have seen for this game is 9/6/5 Double Bonus. I have seen this game and pay table at the $1 level at the Texas Station and the 25 cents, 50 cents, and $1 level at the Plaza. Without the Chase the Royal feature the expected return is 0.978062. The probability of a qualifying pair to switch is 0.097464. The expected value per coin of a jack-king pair is 1.446068. The average expected value of 3 to a royal in chase the royal mode is 1.635714. Overall the player gives up 0.140939 in expected value by giving up the jack-king pairs but gets back 0.159423 in chase the royal mode. The player also gets 0.001847 in expected value from the bonus for having a royal flush on every hand. This is very unlikely to happen unless the player is dealt a pat royal flush and this figure ignores the probability of drawing to a royal flush on every hand (this probability varies depending on the number of hands played, but for the record in 3-play mode only it add another 0.000004 in expected value). The overall return for 9/6/5 Double Bonus is 0.998393.
The following table shows each final hand, the associated probability, and the contribution to the total return. Note that the straight and flush is divided according to whether in or out of chase the royal mode.