Odds

With one exception that we won't worry about for now, we never look beyond the next card.

With that in mind, there are only five winning hands worth considering. All others require one of these as a starting point, or require more than one card.

Typical Hand
J& 9% A% 2* 5*

Obviously, we keep the J& and the A%. But why?

We will be drawing three cards out of the 47 remaining in the deck. Six of those will cause us to win:
J% J^ J* A& A^ A*

Our chance of getting one of these is 6/47. But we're going to have three chances to get one of these!

Here's where my lack of stats knowledge may do me in... but I would venture a guess that the chance of getting one of these six cards over three attempts is about (6*3)/47, or 18/47. That's about a 38% chance... not good, but it's clearly the best chance.

The Basic Premise

While standing in front of the Great Quartersucker, you can't be worrying about dividing by 47s, or even 52s. So I ignore the denominator in all future calculations! The difference between 1/52 and 1/47 is negligible -- 0.2% -- so I have decided it's not worth my consideration.

Those with a deeper knowledge of statistics are welcome to dispute this blanket assertion. On the other hand, anyone who has a grasp of statistics knows already that you can't beat the machine -- if there were a way to beat the poker machine on a regular basis, then the machine would serve no purpose to its owner.

This page will never help you win money. If it helps you lose less money, more slowly, then it has served its purpose.

Comparing odds

Here's a judgement call situation. I've omitted the suit because it is immaterial if there are fewer than three cards of any one suit.

Hand: 2 2 J Q 8

One pair, but a worthless pair on a Jacks Or Better machine. The tendency is to keep the J and the Q. But is that the best plan?

Keep: J Q
Winning cards: J J J Q Q Q (6)
Chances: 3
Odds: 6*3 = 18

What if you keep the pair and try for three of a kind?

Keep: 2 2
Winning cards: 2 2 (2)
Chances: 3
Odds: 2*3 = 6

Ok, the odds suck for keeping the pair vs. the J and Q. But there's another winning combination... Two Pair.

Keep: 2 2 8
Winning cards: 2 2 8 8 8 (5)
Chances: 2
Odds: 2*5 = 10

Still doesn't look as good, does it? And why keep the 8 instead of the J or Q? That leads us to the next addition to the plan!

Weighting for Return

Jacks Or Better only pays out what you paid in. But Two Pair pays back double, and Three Of A Kind pays back 3x. Over time, you only need to get 1/3 as many Three Of A Kinds to get your money back. Let's apply that to the hand above.

Hand: 2 2 J Q 8

Keep: J Q
Winning cards: J J J Q Q Q (6)
Chances: 3
Odds: 6*3 = 18
Return: 1x (Jacks Or Better)
Weighted Odds: 18*1 = 18

Keep: 2 2
Winning cards: 2 2 (2)
Chances: 3
Odds: 2*3 = 6
Return: 3x (Three Of A Kind)
Weighted Odds: 6*3 = 18

The weighted odds for these two options are identical! That was my first big surprise when working out this method. But there's more.

Keep: 2 2 8

Winning cards: 2 2 (2)
Chances: 2
Odds: 2*2 = 4
Return: 3x (Three Of A Kind)
Weighted Odds: 4*3 = 12

Winning cards: 8 8 8 (3)
Chances: 2
Odds: 3*2 = 6
Return: 2x (Two Pair)
Weighted Odds: 6*2 = 12

Combined Weighted Odds: 12+12 = 24

Wow! By keeping the existing pair plus one card, we have increased our odds by 6 points over either keeping just the pair or keeping just the royalty. And it doesn't matter what the other card is! In real poker, an Ace High Two Pair beats a Trey High Two Pair. On the machine, it doesn't matter if it's deuces or aces in this game... Two Pair is Two Pair. Pick the singlet that tickles your fancy, royalty notwithstanding.

Now for the gut check.

Hand: 8 8 J Q A

Yikes! We've got three kinds of Jacks Or Better combinations! How does that affect the calculation?

Keep: J Q A
Winning cards: J J J Q Q Q A A A (9)
Chances: 2
Odds: 9*2 = 18
Return: 1x (Jacks Or Better)
Weighted Odds: 18*1 = 18

Keep: 8 8 J
Combined Weighted Odds: 12+12 = 24

It's still better to keep the pair... believe it or don't.

Straights

Consider this hand.

Hand: 5 6 7 8 A

You'd almost certainly keep the 4/5 of a straight, but you might hate to see the Ace go away. Let's calculate the odds.

Keep: A
Winning cards: A A A (3)
Chances: 4
Odds: 3*4 = 12
Return: 1x (Jacks Or Better)
Weighted Odds: 12x1 = 12

Keep: 5 6 7 8
Winning cards: 4 4 4 4 9 9 9 9 (8)
Chances: 1
Odds: 8*1 = 8
Return: 4x (Straight)
Weighted Odds: 8*4 = 32

Wow! That's a huge difference! Ok, we've proven what we knew by instinct.

But what about this hand?

Hand: 5 6 8 9 A

That's an inside Straight, and everyone knows you don't deal to an inside Straight. Or do you?

Keep: A
(Calculations already performed above)
Weighted Odds: 12x1 = 12

Keep: 5 6 8 9
Winning cards: 7 7 7 7 (4)
Chances: 1
Odds: 4*1 = 4
Return: 4x (Straight)
Weighted Odds: 4*4 = 16

The chances are half as good as in an outside Straight, but still better than keeping the Ace.

Now, what about this?

Hand: 5 6 8 9 8

Do you go for the straight, or keep the pair and a singlet?

Keep: 5 6 8 9
Weighted Odds: 4*4 = 16

Keep: 5 8 8
(See "Weighting For Return" for calculations)
Combined Weighted Odds: 12+12 = 24

Verdict for the plantiff: Keep the pair and a partner (24) over an inside Straight (16).

But with the 32 odds for an outside Straight, drop the pair and go for the Straight.

Tricky Straights

Straights can include royalty. This increases your odds... but it makes your choices more difficult. You've also got half of Jacks Or Better!

Hand: 8 9 10 J 2

Keep: 8 9 10 J

Winning cards: 7 7 7 7 Q Q Q Q (8)
Chances: 1
Odds: 8*1 = 8
Return: 4x (Straight)
Weighted Odds: 8*4 = 32

Winning cards: J J J (3)
Chances: 1
Odds: 3*1 = 3
Return: 1x (Jacks Or Better)
Weighted Odds: 3*1 = 3

Combined Weighted Odds: 32+3 = 35

Great! Does it hold for an inside Straight?

Hand: 7 9 10 J 2

Keep: 7 9 10 J

Winning cards: 8 8 8 8 Q Q Q Q (8)
(Remember how we calculated the odds for the inside Straight?)
Weighted Odds: 4*4 = 16

Winning cards: J J J (3)
Weighted Odds: 3*1 = 3

Combined Weighted Odds: 16+3 = 19

Easy enough! We can even figure out our best choice when we have a couple of face cards.

Hand: 8 10 J Q 2

Keep: 8 10 J Q
(we've got J J J and Q Q Q helping us out now)
Combined Weighted Odds: 16+3+3 = 22

Keep: J Q
(This was the first example!)
Weighted Odds: 18*1 = 18

That was close! If it weren't for the effect of keeping the J and Q, we would have been forced to try for Jacks Or Better. Now, let's turn up the angst a notch.

Hand: 9 10 J Q A

Inside Straight, outside Straight, or Jacks Or Better? Well, we've got some of the numbers...

Keep: J Q A
(three chances at Jacks Or Better)
Weighted Odds: 18*1 = 18

Keep: 9 10 J Q
(outside Straight plus two Jacks Or Better)
Combined Weighted Odds: 32+3+3 = 38

Keep: 10 J Q A
(inside Straight plus three Jacks Or Better)
Combined Weighted Odds: 16+3+3+3 = 25

Outside straight comes out the winner, as much as you hate to lose your Ace.

Flushes

Flushes are the surprise bugaboo of the process. Consider that you have a 1 in 4 chance of drawing card of a particular suit -- in our parlance, that's a 13 chance. On top of that, Flushes have the highest multiplier of all the hands we're considering -- five times your investment.

So the simplest rule is: if you have 4/5 of a Flush, Flush it!

Hand: ^ ^ ^ ^ %

Keep: ^ ^ ^ ^

Winning cards: ^^^^^^^^^ (9)
Chances: 1
Odds: 9*1 = 9
Return: 5x (Flush)
Weighted Odds: 9*5 = 45

Nothing we've calculated before comes close to this 45 chance! The highest non-flush chance would be this hand...

Hand: J Q K A 4
Keep: J Q K A
(outside Straight plus 4 Jacks Or Better)
Combined Weighted Odds: 16+3+3+3+3 = 28

Oops! Until they let you wrap a straight K-A-2, that's an inside Straight hand -- only 10 gets you a straight. Let's try again, but this time it's an outside Straight vs. a Flush.

Hand: 10^ J^ Q% K^ 4^

Keep: 10 J Q K
(outside Straight plus three Jacks Or Better)
Combined Weighted Odds: 32+3+3+3 = 41

Keep: 10^ J^ K^ 4^
(Flush plus two Jacks Or Better -- gotcha!)
Combined Weighted Odds: 45+3+3 = 51

Still not as good as a possible flush. And if you've got three face cards of the same suit, your weighted odds are 54. And if you've got four...

Hand: 8% J^ Q^ K^ A^

... let's just say that the multiplier for a Royal Flush is "significant."

One more Flush call. You'll love this one.

Hand: 3^ 4^ 9^ J^ J%

Obviously, it's a Flush with one Royal vs. a Pair.

Keep: J J 9
Combined Weighted Odds: 12+12 = 24

Keep: 3^ 4^ 9^ J^
Weighted Odds (Flush): 9*5 = 45
Weighted Odds (Jacks Or Better with only 2 available): 2*1 = 2
Combined Weighted Odds: 45+2 = 47

Clearly, the chances on the Flush are much better -- almost twice as good with weighting. So what if you had Jacks Or Better already and have to give it away for a chance on a Flush? That's why they call it "Gambling!"

Flush Twice: The Exception

Every rule has an exception. The nearly 1 in 4 chance of getting a card of a particular suit is high enough to warrant breaking the hard and fast "don't look past the next card" rule.

I've already told you that I don't know Statistics. But I remember just enough to be dangerous. I remember that the chance of two events occuring in sequence can be calculated by multiplying their odds together. So let's say I want the odds for drawing a 2^ and a 3^ off of a new deck...

1/52 * 1/52 = 1/2704 = 0.0%

That's a tiny number! Even something more likely -- any 2 and any 3 -- is pretty unlikely.

1/13 * 1/13 = 1/169 = 0.6%

While that's greater than the 0.2% difference between 1/52 and 1/47 that I chose to ignore, it's still pretty tiny -- one point under my method is worth about 2.0% before multiplying by return.

But what are the chances of drawing two hearts?

1/4 * 1/4 = 1/16 = 6.25%

That is a non-trivial value! A 1 out of 16 chance is equivalent to 3.25 out of 52 -- you have a better chance drawing two hearts than you do drawing any one of the first three clubs.

In fact, I could just plug 3.25 into my point model, but I don't want to deal with quarters. Do I go down or up? And is my assumption even rational? For this, I'll try to calculate the actual odds.

Hand: ^ ^ ^ % &
Winning cards: 10
Cards remaining: 47

Here's where a stats whiz would be helpful! I assume I've got to remove the matching ^ from the numerator and denominator before calculating the final odds.

10/47 * 9/46 = 90/2162 = 4.1%

That is a good bit lower than my original 6.25% Why? Because 10/47 = 21.3% and 9/46 = 13.0% -- both much lower than 25%.

4.1% of the 52-card deck is 2 cards. This may be a bit low, since the other scores might also be lowered if subjected to this sort of analysis... so I'll bump it up to 3, which is close to the 3.25 I originally estimated.

Does it help?

Hand: 2^ 4^ 7^ J% Q%

Keep: J Q
Winning cards: J J J Q Q Q (6)
Chances: 3
Odds: 6*3 = 18
Return: 1x (Jacks Or Better)
Weighted Odds: 18*1 = 18

Keep: ^ ^ ^
Odds: 3 (special)
Return: 5x (Flush)
Weighted Odds: 3*5 = 15

Even with the big return on a Flush, it can't beat the Weighted Odds up above Jacks Or Better with two cards. The same applies to three:

Keep: J Q A
Winning cards: J J J Q Q Q A A A (9)
Chances: 2
Odds: 9*2 = 18
Return: 1x (Jacks Or Better)
Weighted Odds: 18*1 = 18

Oh, yeah... the chances on 3 face cards are the same as on 2 face cards. Flush again!

But the chance with a single face card has a different result:

Hand: 2^ 4^ 7^ A% 9%

Keep: A
Winning cards: A A A (3)
Chances: 4
Odds: 3*4 = 12
Return: 1x (Jacks Or Better)
Weighted Odds: 12x1 = 12

The possible flush's 15 beats the single face card 12.

Conclusion

Links

A while back, I did a search on "Video Poker Cheat Sheet" and came up with very little useful content. Fortunately, things have changed! These links look decent -- but caveat emptor. That's Latin for, Don't blame me if these links make you lose even more at the Video Poker machine!