# A day full of Royals

April 10, 2018 3:09 AM
by

I had a truly extraordinary day playing video poker the other day.

I was playing my favorite and obscenely volatile 5-play multi-strike machine. It was not going well initially, when I hit a Royal from a 3-Card Royal on the second line. Any day I hit a Royal is an extraordinary day. Any day just about anyone hits a Royal is an extraordinary day.

At roughly 1 in 40,000 hands, it takes a lot of hours on any type of machine (except maybe 100-play) to hit one. Based on the total number of video poker hands I have played in the past two plus decades and the number of Royals I’ve had, I think I’m still owed about a dozen!

The last two I’ve hit were both from a 3-Card Royal. The one before that was from a 4-Card Royal. When you hit it in from the latter, you’re excited but you almost expect to do it. When you hit one from the former, it’s a complete shock. It’s a 1 in 1,081 shot. That’s 1,080 misses for every one hit. But that hit is so sweet. Hitting it from a 4-Card Royal is “only” 1 in 47. So, how strange is it that my last two came from a 3-Card Royal? It’s not all that strange at all.

To better understand this, we need to take into account how often you are dealt a 4-Card Royal vs. a 3-Card Royal. You’ll be dealt a 4-Card Royal 936 times out of the 2,598,960 total possible 5-card deals. This is 0.036%, or just under 1 in 2,800 hands. We multiply this by the 1 in 47 chance of getting the fifth card, and we come to a final probability of 0.000766% or 1 in 130,500-plus hands.

By contrast, a 3-Card Royal will be dealt to you (and played) just over 1% of the time. When we multiply this by the 1 in 1,081 chance of hitting the Royal, we get a final frequency of 0.001009%, or 1 in about 99,000 hands. Seeing this, we can understand why it is not such a surprised that two of my last three Royals have come from a 3-Card Royal.

If we continue this to a 2-Card Royal, we find we will play a 2-Card Royal 7.64% of our hands. Only 1 in 16,215 will turn out to be a Royal. In the end, we find 1 in 212,000-plus of our total hands will be a 2-Card Royal that turns into an actual Royal. To the best of my recollection, I’ve never had one of these and this should not be much of a surprise.

What made my day truly extraordinary was not the Royal, however – despite what I said earlier. It was the Quad Aces. I got them seven times over a 2-plus hour session. I don’t really know how many hands I played in that time. Given the nature of a 5-play multi-strike game, it is a bit hard to even estimate. But, in 2-plus hours, it wasn’t tens of thousands of hands. I wasn’t keeping track of all my Quads, but there is a possibility I had more Quad Aces than all other Quads put together. At one point, I hit them on two consecutive deals. In the case of Multi-Strike, a deal is a whole new game (i.e. more money wagered).

What I found curious about these Quads, a bit like my Royal, is I hit five of them from a Pair, one from Trips and one from a Single Ace (which was actually an off-suit AK I held). I suspect we’ll find the probabilities mimic the pattern we saw from the Royals. While you are far less likely to hit Quads from a Pair than you are from Trips, the fact you are dealt a Pair way more often means overall this is how you’ll get most of your Quads.

You’ll be dealt Trips about 2.1% of your hands. You’ll get the Quads from them 4.25% of the time. The overall is about 1 in 1,112 hands dealt will be Trips turning into Quads. You’ll be dealt Pairs a whopping 41.2% of the time. You’ll turn them into Quads about 0.11% of the time. When we do the math, we find 1 in 875 deals will be a Pair turning into Quads.

So, getting five out of my seven in this manner was a bit “off,” but hardly astronomical. There was no way for me to keep track, but if anything was off it was how often I hit the Quads from a Pair of Aces on this particular day. What should’ve been 1 in 360 seemed way more often yesterday.

As for my hold A-K and draw three Aces. Well, that was a one in 16,215, so I won’t be expecting that to happen again anytime soon!