Interesting article about Royal Flushes vs. K high straight flush

I came across this page recently which maintains that “you are more likely to get a Royal Flush than a King-high Straight Flush”.

This naturally seemed intriguing and worth a read. I remain unconvinced however and wondered if anyone else had thoughts on the math used here and whether it is indeed true.

I lost a little confidence in the article when the author states that " There are 4,234 ways to make a Royal Flush from any seven cards" when earlier the author stated there were 4,324 ways, although this could just be a typo.

Any thoughts welcome about the logic/math involved here - certainly interesting for sure!

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I read the article and I don’t know how I feel about the data. The author seems to be all over the place with employment in life . I read his bio and it seems impressive but at the bottom of the page is the click to buy him a coffee at $3.00 if you liked his article. That has been proven to be a bottom of the food chain company.

As far as his data I think he just jumped in on the poker space because it’s popular. This is not the authors only article on a plethora of topics and in the end of it he’s asking people to buy him a $3.00 coffee. Seems odd when he sold his company to Microsoft.

I’m going to look more into the data on his hypothesis.

This is his current company so I’m a bit skeptical of the whole idea and data.

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Interesting. Its, as he says, the other two cards. For a royal flush the other two cards can be anything. But for king high straight flush the other two cards can be anything but the ace of the flush suit. That much probablity is reduced in making king high straight flush.

In saying “You’re more likely to get a Royal than a K-high SF”, what we’re actually saying is, “If you’ve made a K-High SF, there’s still two chances (your two other cards) of having the Ace, changing your hand to a Royal. But when holding a Royal, there’s ZERO chance of the other two cards changing your hand.”

Of course, I can also see how one would reason the other way: if you have 5 cards that make a 9-K SF, you have (1/47 + 1/46) = 4% chance of drawing the Ace, but you have 1-(1/47 + 1/46) = 96% chance of NOT drawing the Ace, so you are “more likely” to just keep the 9-K SF.

So I guess it depends upon how you do your calculations. No doubt there’s a right way and a wrong way, but not being a mathematician, I’m not sure which is which. Either way, I’m going all-in with it!

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He didn’t really use any math when making his point. It’s written in a clickbait/infotainment style so I didn’t take it seriously at first.

But I got to thinking about it. Imagine an absurd form of poker where instead of being dealt 7 cards you were dealt 52 and required to make the best 5 card hand you could. You would always have a royal flush and the probability of a K high straight flush would be 0%. If it was 48 cards instead, you would always have a royal flush except in exactly one runout where the 4 aces were not dealt to you, in which case you would have a king high straight flush. Quite rare but not zero. But the probability of quads would be zero. If that trend continues as you reduce the number of cards in a players hand, there should still be a lot of cases when more than 5 cards are dealt where weaker hands are “counterfeited” by stronger cards, making them less likely.

Intuitively it feels wrong but I don’t know why it isn’t. There are exactly 4x(47x46/2)=4234 seven card hands that contain the cards A-T of the same suit. There are 4x(46x45/2)=4150 seven card hands that contain K-9 of the same suit without having the relevant ace. So there are 184 fewer strictly K high SF combos? This makes a royal flush actually the most likely straight flush to make in a 7 card game, since it can’t be “poisoned” as the author put it. Because this makes the Q high SF even rarer, because it can be “poisoned” by either the presence of a suited K or AK in the other two cards. I often make the mistake of double counting combinations into multiple categories when thinking about poker hands, my gut tells me I’ve done the same here but I don’t see how.

I had now an App running, it searched in all 7 of 52 combinations for Straight/Royal Flushs, and the result was this:

SF

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The odds of any exact set of 5 cards is equal to the odds of any other exact set of five cards.

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Exactly!

Interesting, but not sure if I could validate. Checking my best hands, I have 9 royals & 1 King high straight flushes and only allowed to see top 10. But I would be happy to get any straight flush. Shooting for number 10, GL to everyone at the tables & Have Lot’s of fun

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