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Any video poker player who has put in even a little time playing the game has been dealt four cards of a royal flush. When this happens the player will carefully select each of the four cards to be saved - sometimes even allowing the fifth card of a flush to be discarded. Then, while holding their breath, the "deal" button is pressed. Most often disappointment is the result. No royal flush, no flush, and no straight. Usually not even a high pair.

This same scenario can happen over and over - five times, 10 times, 20 times or more. Surely the video poker games must be rigged. Certainly one time out of twenty should produce a royal flush. After all, it is rare to be dealt four of a royal flush.

I have heard this complaint from many people, some of whom I consider friends and savvy players. It seems players are always looking for proof that the game they love to play is rigged. In an effort to dispel the paranoia, let's take a look at the odds of completing several of those hands.

Let's start with the four cards of a royal flush. Yes, it is rare to be dealt four of a royal. The odds of this happening are about 1 in 2,765 or roughly once every three to six hours of play depending on your speed.

Four of a royal flush is a very powerful hand, since completing the royal pays 800-for-1 (4,000 for 5 credits played). In a Jacks or Better game, saving four of a royal flush comes right after saving four-of-a-kind or better.

After your initial five cards are dealt, there are 47 cards left in the deck. Only one of these remaining cards will complete the royal so you have a one in 47 chance of completing the jackpot hand. Keep in mind that this is an average over time. It could take 200 or even 300 times being dealt four of a royal before you complete one. Of course it could also happen immediately and even two or three times in a row, but on average you will complete a royal flush from four cards of a royal once every 47 times. It takes a player playing 600 hands per hour almost 217 hours - on average - to play enough hands to complete one four card royal.

Of course not every failure to produce a royal flush ends with zero money won.

There is a 1 in 47 chance of a straight flush if the ace is the only missing card of the royal (the suited 9).

In the remaining 47 cards there are eight or nine cards of the same suit (depending on whether a card of the same suit was discarded in pursuit of the royal) that will complete a flush - about a 1 in 5 chance if a like suit was not discarded and just under a 1 in 6 chance if the same suit was discarded.

Assuming the fifth card of a straight was not discarded in the attempt to produce a royal, there is about a 1 in 8 shot of completing a straight. If it was discarded there is still about a 1 in 9 shot of scoring a straight.

There is also the possibility of snagging a high pair when saving four of a royal. If the 10 is not one of the saved cards, there are 12 cards in the 47 that will match one of the saved cards for a little better than a 1 in 4 shot at getting your bet back. If the 10 was saved there are only nine possibly high pair matches remaining for a little worse than 1 in 5 odds of returning your bet.

That is how four of a royal flush plays out. How about the odds of completing other four card hands such as four of a straight flush, four of a flush, four of a straight, or two pair making a full house?

The odds of completing an inside (meaning when there is a gap in the straight) straight flush when dealt four are the same as completing a straight flush when dealt four of a royal since only one card will complete the desired hand. If the four cards dealt are all in sequence with no gaps (this is called an open straight or an open straight flush) the odds of completing the straight flush are 2 in 47. If there is a gap, the odds are 1 in 47.

Saving four of a straight flush is also very high in the strategy chart - just behind 3-of-a-kind or better for an open straight flush and just two lines lower (after saving two pairs) for an inside 4-card straight flush.

When it comes to four of a flush, there are nine cards that will complete the flush in the 47 remaining cards - or about a 1 in 5 chance of snagging a flush.

Like the straight flush, the odds of drawing the correct card to complete a straight depends on whether the cards are together (open) or have a gap (inside). If it is four cards of an open straight, there are eight cards out of 47 that complete the straight for just under a 1 in 6 chance. When there is a gap (or the ace - either high or low - is one of the cards), the odds of completing the hand are cut in half as there are only four cards that will do the trick. There is a 1 in almost 12 chance of completing the hand in this case.

The last four-card dealt hand I will review is two pair. The best possible outcome when holding two pair is a full house. Since there are two more of each of the two pair's rank in the remaining 47 cards, there is a 1 in almost 12 chance of completing the full house - the same as completing four of an inside straight.

How about being dealt 3-of-a-kind? What are the odds of scoring a quad?

On average a 3-of-a-kind hand is dealt about every 47 hands. When this happens, there is exactly one card in the remaining 47 that will complete the 4-of-a-kind. However, there are two shots of getting the winning card since two cards have been discarded. Therefore there is about a one in 23 chance of completing a 4-of-a-kind when dealt 3-of-a-kind. This may not sound very high, but on average you will complete a quad from trips only about once every two hours or so.

While being dealt four of any potential high paying hand is an exciting proposition, actually completing the hand is still a far from common event. Don't let your hopes and expectations blind you to the reality of the math. The math will win out in the end.

This article showed the odds of completing various combinations of four card hands. Let's look at the strategy for playing some hands that fall into those categories. Playing full pay Jacks or better with 5 credits played, you are dealt the following hand: (the meaning of the subsets is: _s - Spades, _c - Clubs, _d - Diamonds, _h - Hearts.)

A_s J_h Q_d 9_c T_h

How would you play it?

You could save the Ace, Jack, Queen and Ten going for a high pair, two pair, three of a kind, or straight. You could save the Ace, Jack and Queen going for a high pair, two pair, three of a kind, or straight. You could hold the Queen and Jack going for all the previous as well as a possible four of a kind. You could hold the Jack and Ten going for a high pair, two pair, three of a kind, straight, flush, four of a kind, or a royal flush. You could hold the lone Ace, Jack or Queen and go for all of the previous as well as a straight flush.

The best hold however is the Jack, Queen, Nine and Ten as this is an open straight.

Returns for each mentioned save are as follows.

Jack, Queen, Nine and Ten - 4.04 credits

Ace, Jack, Queen and Ten - 2.66 credits - nearly half the optimum hold

Jack and Queen - 2.41 credits

Jack and Ten - 2.33 credits

Ace and Queen - or - Ace and Jack - 2.32 credits

Ace only - 2.24 credits

Queen only - 2.23 credits

Ace, Jack, and Queen - 2.21 credits

Jack only - 2.20 credits

Here is another - also full pay Jacks or Better with five credits played.

You are dealt:

A_d 7_c 5_s 4_s 6_s

How would you play this one?

The only viable options are saving the Ace, or saving the three of an open straight flush (5_s 4_s 6_s) or saving four of an open straight (7_c 5_s 4_s 6_s).

Saving the Ace returns 2.38 credits on average.

Saving three of an open straight flush returns 3.02 credits on average.

Saving four of an open straight returns 3.40 credits on average and is the proper save in this example.