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No sooner did I write last week’s column about how frustrating Three of a Kinds can be when they don’t turn into Quads, when I had the pleasure of playing for a bit this weekend and proving it.

Playing Multi-Play Multi-Strike, I was dealt Three Aces on three different occasions. Each time, I come up empty. That’s 15 shots at pulling that fourth Ace. The odds were against me getting that ace, but just barely. Roughly 52% of the time, with 15 shots at the Ace, I won’t hit it. That leaves 48% of the time, I would hit one or more! It is a $20 quad hand. Not exactly life changing, but session changing. It would have been a nice win even on the bottom level.

But to show how funny life or at least how funny video poker can be, a short time later while playing three hands on the top line, I was dealt a Queen and four useless cards. Not exactly the hand you want when you’re being paid 8x the usual payout. I held the Queen and hoped.

On one of the hands, I was dealt Quads! But, not of Queens. I was dealt four 9’s to go with the held Queen. This was a mere 22,295-to-1 shot. The odds of being dealt a Four of a Kind off the deal is only one in 4,165. This paid the same $20 the four Aces would’ve paid if I had hit it with a kicker on the first line.

I’ve hit Quads from a single high card before, although I’m not sure if I’ve ever hit it by being dealt four of a different rank from the one I’ve held. This led me to wonder how often you get Quads from each different type of draw.

As I’ve written before, the most likely way to get a Royal Flush is being dealt a 3-Card Royal and then picking up the other two. Yes, if you are dealt a 4-Card Royal, you are more likely to hit the Royal than being dealt a 3-Card Royal. But, you’re dealt 3-Card Royals way more often!

So, what happens with Quads?

There are two pieces of information we need to know. The first is how often we get a certain number of a card on the deal; we multiply that by the probability of getting to Quads. So, let’s start with the easiest one. If you are dealt Quads, you have Quads and you are done. The probability of being dealt quads is 0.024%. One down.

Next up, you might be dealt Three of a Kind and draw the fourth one. You’ll be dealt Trips about 2.1% of the time and get Quads from it 4.26%. When we multiply this together we get about 0.09%. Not surprisingly, this is about 3x-4x as likely as getting dealt Quads.

We get a lot of Pairs; 41.2% of our hands will be Pairs that we play. We sometimes discard a Low Pair for a 4-Card Flush or a High Pair for a 4-Card Royal. You’ll turn these Pairs into Quads a mere, 0.2775% of the time. That’s only one in 360. But, overall, 0.1143% of our hands will start as a Pair and become Quads. This is still a bit more likely than starting with Trips and getting Quads.

Next up is the single High Card. From a single High Card we have two ways of getting Quads. The first is to draw three to the rank held and the other is the way I got it this past week, where I draw four of a single rank. We’re going to put this latter group into our final grouping in a moment where we start with zero and draw four. We get a lot of High Card hands, too.

Roughly 15.5% of our hands will be played as a single High Card. Only one in just over 4,000 will turn into Quads by drawing three to the rank we hold. This is 0.0038%. You are more likely to be dealt Quads than a single High Card and draw three to it. This number is reduced because so many of our High Card hands are played as something else.

Last but not least, we have two ways to start with zero cards of a particular rank and draw four. We can do this by drawing Quads from a Razgu or by the way I did it earlier this week, drawing 4 of a Kind after holding one High Card (of a different rank from the Quads). Doing it this latter way will happen only 0.00069% of the time. This is one in 144,000 overall.

So, while I had a 22,295-to-1 shot of drawing Quads once I held the one High Card, the overall probability of the event occurring was one in 144,000! Starting with a Razgu, the odds of drawing Quads is one in 137,377 – a smidge more likely than what happened to me. Overall, the odds are not strong in getting Quads this way. It works out to be a mere 0.0014%. This is a bit more than half as likely as being dealt Quads in the first place.

So, we see that pattern for being dealt Quads is a lot like the one for being dealt a Royal Flush. The overwhelming majority come from being dealt Trips or a Pair, just like most Royals come from 4-Card Royals and 3-Card Royals. When we take a closer look we also find more Quads will come from a Pair than Trips, just as more Royals come from 3-Card than four.

About the Author

Elliot Frome

Elliot Frome’s roots run deep into gaming theory and analysis. His father, Lenny, was a pioneer in developing video poker strategy in the 1980s and is credited with raising its popularity to dizzying heights. Elliot is a second generation gaming author and analyst with nearly 20 years of programming experience.

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