For the average player (as well as the not so average player) hitting a royal flush at video poker is a rare and exciting event. Years can pass between them. Just what are the odds against hitting a royal flush? How about the odds against getting royal flushes back to back?
Let's investigate those questions.
Contents
- A royal flush is usually treated differently by the casino than other winning hands
- The playing strategy matters
- The odds against getting a royal flush immediately after a previous royal flush
- The odds against being dealt two royal flushes in a row
- Summary
A royal flush is usually treated differently by the casino than other winning hands
Although there are a few video poker casino games and pay tables that have winning hands that can be as rewarding to the player, on most games, a royal flush pays more than any other hand.
Depending on the denomination the game being played, the payoff meets or exceeds the IRS limit for a win ($1,200) and a W2-G form must be generated. This requires casino personnel to get involved. The machine locks up and plays a happy sounding melody while help is summoned. If it is a busy time, the music can get on your nerves during the long wait.
Once the attendant arrives, he or she verifies everything is proper, gets your personal tax information, and heads back to get your hand pay and W2-G form.
Usually two attendants return – one with the cash and tax form and the other as a witness. Once you are paid, the machine is unlocked and you are asked to play off that glorious royal flush.
It is bad for business to have a video poker machine showing the last hand as a royal flush. Nobody wants to play a game that has just scored a hand that occurs only once in about 40,000 to 45,000 hands or so. Players would much rather sit at a game where a royal has not appeared in a good long time.
Players who have just collected for a royal flush generally do not mind playing off the royal. They have just received a very nice pile of cash and are feeling flush.
Once curiosity got the best of me, however. After getting a royal flush, getting my W2-G, and my winnings, I was asked to play off the royal flush. My response was, what if I refuse to do so? The attendant was speechless for a few moments. The initial response was something to the effect of back-to-back royal flushes have been known to happen.
“Have you ever seen it happen,” I asked. “No,” was the response, “but I know it has happened.” I do not remember exactly what else was said, but after a brief while, and to the apparent relief of the attendant, I played off the royal flush.
I know other players who got royals within a few hands of each other. In fact, I am one of them. This was not one of those times that back-to-back royal flushes happened.
The question (and topic of this article) is, if a royal flush occurs once in about 40,000 to 45,000 hands, what are the odds against getting two of them back-to-back?
The Playing Strategy Matters
The odds vary according the specific game and pay table as well as the strategy employed while playing the game. The normal video poker playing strategy has the player maximize the return of each hand being dealt – not maximizing the chance of getting a royal flush.
By maximizing return, the player has the best overall chance of winning. This strategy also allows the player to play for the longest possible time on a particular bankroll.
If the playing strategy was tilted toward getting royal flushes, they would happen more frequently, but the player would also throw away hands that would produce other, less lucrative winning hands. The net result is the player would lose more by chasing royal flushes.
On a standard full-pay jacks or better game, a good example of holding for the maximum return of each hand rather than for a royal is a dealt hand containing a suited ace-10 with mixed junk cards. The correct hold for maximum return is to hold the lone ace. This hold guarantees the player will NOT get a royal flush, but in the long run will win more money.
Each different game can affect the playing strategy. Each different pay table can affect the playing strategy. Finally, the playing strategy affects the frequency of royal flushes.
This article uses only the standard full-pay jacks or better game strategy for its calculations.
Odds Against Getting a Royal Flush Immediately After a Previous Royal Flush
Keeping in mind that we are using a strategy that favors a maximum overall return, a standard full-pay (9-for-1 for a full house, 6-for-1 for a flush) jacks or better game strategy will see a royal flush occur once every 40,390 hands, on average. This a frequency of 0.0024758%.
While the player may very infrequently be dealt a royal flush, the frequency numbers are based on the player having the opportunity to select replacement cards before the final hand is determined.
If I have done my research correctly, the frequency of back-to-back royal flushes is 0.0024758% times 0.0024758% (0.000024758 x 0.000024758) which equals 0.0000000612988%. Put another way, this will happen, on average, once every approximately 1.6 billion hands. For comparison, the odds against being hit by lightning in any given year are about a million to one.
Odds Against Being Dealt Two Royal Flushes in a Row
The odds against having two royal flushes in a row are, indeed, long. How about the odds against being dealt a second royal flush immediately after being dealt the first one?
On average, a royal flush is dealt once every 649,740 hands. This works out to a frequency of 0.0001539%. Using the same formula, the frequency percent of two dealt royal flushes in a row 0.0001539% times 0.0001539 %.
This equals 0.0000000002369% – once every approximately 422 billion hands. Talk about long odds!
Summary
- Hitting a royal flush in video poker is a rare, exciting, and lucrative video poker event.
- Casinos want players to play off a royal flush because leaving it showing would cause other players to avoid playing that machine.
- It is stated that hitting two royals in a row has happened – but what are the odds?
- While hitting a royal flush is rare, hitting two in a row happens only once every approximately 1.6 billion hands, on average.
- Hitting two dealt royal flushes in a row is ridiculously rare – once every approximately 422 billion hands, on average.
- The odds against getting hit by lightning in a given year is “only” about a million to one.
