I Want To Win $100,000: What Are My Odds?

There is a video poker machine at my local casino. The usual, you get 5 cards and can discard 0-5 cards for additional cards.

On one of the machines, it will pay out $100,000 but ONLY if you get the Royal Flush in the exact, left to right order (any suit):

10 Jack Queen King Ace

  1. What are the odds of getting that hand, in that exact order?

And as long as I am asking, there is another machine that will pay you $12,500 if you get the Royal Flush in either left to right or right to left:

10 Jack Queen King Ace (or) Ace King Queen Jack 10

  1. What are the odds of getting either one of those hands?

You have 52 possibilities for your left-most card, 51 for the second, 50 for the third, 49 for the fourth and 48 for the right-most card. This gives you
5251504948 = 311,875,200
possibilities. Exactly four of those are Royal Flushes in the correct left-to-right order, and another four are Royal Flushes in the correct right-to-left order. So your chances of winning $100,000 are 4/311,875,200 = 1 in 77,968,800, and of getting the $12,500 prize 8/311,875,200 = 1 in 38,984,400.

FYI: The chances to be dealt a Royal Flush in any order is 4/2,598,960 = 1 in 649,740.

But, Schnitte, those are odds for being dealt a sequential royal first time around. DMark can improve his hand by drawing to it, so the odds are better than 1-in-77.9 million.

I’ll try some heavy-duty wild guessing here. Frank Scobelete claims the odds for drawing to any royal are 1-in-40,000. The odds of that royal being sequential, as I see it, are 5 factorial. (Take a 10-J-Q-K-A out of the deck…the odds of getting the sequential should be 5!). 5! times 40,000 is 4.8 million. So, roughly speaking, the odds of the sequential royal are 1-in-4.8 million. The odds of getting the reverse sequential are also 1-in-4.8 million, so the two together have odds of 1-in-2.4 million.

Sounds like a sucker bet to me. I assume that it’s a 25-cent machine, and to hit the jackpot you have to play full-coin, or $1.25 a play. The “true” payoff for the sequential should thus be $6 million. Instead, it’s $100,000. And you wonder why the casinos are so big and plush.

Yes, it is a quarter machine, and yes, you do have to play $1.25.

I kinda figured the odds wouldn’t be all that great…still, I might as well continue to play at that machine…odds of getting a Royal are the same as any other, and on that off chance I would actually get a sequential Royal…

Thanks for the informative, if somewhat disheartening, news!

The odds of getting a royal may be the same, but is the payoff the same as a machine without the sequential bonus? If it’s not, you’re probably better off playing the other machines.

Based on the odds, I’d say you might as well tell the casino to take their video poker and shove it.

What’s the payoff for a non-sequential royal flush? Usually on these machines, the more common, lesser payoffs are made lower than normal to balance out the bigger, top-level pay-out. But if that’s not case, and the other payouts are in line with a decent video poker machine, there’s no reason not to keep playing it.

Usually the payouts for a royal, at least on the decent machines, are 4000 units (assuming you’re playing full-coin) or higher. I’ve seen 5000-unit and progressive machines starting at 4000 units. So, if you’re playing a quarter machine, it’s a $1,000 or up payoff.

In Vegas, there’s a fairly unique royal payoff of $1,199 on quarter machines. The reason for this strange number? Video poker and slot wins of less than $1,200 are not subject to tax.

The SDMB tax superhero must set someone straight again:

Wrong-o, Duke.

It may be the case that slot wins of less than $1,200 don’t have to be reported directly to the IRS by the casino (via a 1099 or W-2G or something), but the money is most definitely and uncontroversially no-two-ways-about-it 100% taxable income to the lucky recipient.

Hm. Yep. Right. Actually I totally forgot about this possibility, but since I wouldn’t have known how to compute it anyway, doesn’t matter.
In any case the $100,000 prize, although it sounds great, is far lower than it ought to be judging from the odds of winning it.

Question: what is the convention on ordering original vs. replacement cards? In other words, if my original hand is:


and I turn in the 8, 2, and 6, and get Q-K-A in return, is the final hand:

10-J-Q-K-A (hurray!)

where the jack has indexed over to the second spot, and the replacement cards are placed in the right-most slots, or is the final hand:

10-Q-K-J-A (boo!)

where the jack stays in position four, and the replacement cards are inserted interstitially?

On a standard video poker machine, the cards stay where they are. And since I’m sure they obviously don’t want to give you any help to win the jackpot, I’m sure that convention remains on this machine!

For each of the four suits, the odds of being dealt all 5 cards correctly were already dealt with by Schnitte - 1 / (52 P 5) = 1/311875200.

So what are the odds of being dealt 10C JC QC KC ??, where the ?? is anything but AC, and the drawing AC? It’s the same. Because the odds that the 1st, 2nd, 3rd, 4th, and 5th cards in the deck form this hand is the same as if the 1st, 2nd, 3rd, 4th, and 6th cards do. In fact, it’s just the same for any set of cards that you’d want to draw. For instance, the following two deck configurations are equally likely, and each one will lead to the same final hand:

10C JC QC KC AC ?? ?? ??
10C ?? ?? KC ?? JC QC AC

You’re just as likely to be dealt a Royal Flush originally as you are to be dealt the 10 and K and then draw the J Q A, or be dealt the 10 Q K A and draw the J, or be dealt nothing, and draw all five. (If the cards get shifted down as zut mentions, it doesn’t change the odds at all, just the workable configurations.)

So now we need to count the number of possible card-draws there are. You can either draw or not draw your first card, draw or not draw your second card, and so on. So the total is 5[sup]2[/sup] = 32. Thus the total probability is 32 times the probability of getting it all on the first draw, or 32 / 77968800 = 1 / 2436525, which matches Duke’s number.

(Now of course nobody plays like this. If you were dealt 10H 10C 10S 10D 5D, would you throw away the last four cards and keep the 10H? I didn’t think so.)

This is only true if the sequential Royal Flush is the only hand that pays. In order to determine how skewed the machine is, you have to look at the expected payoffs for all the different hands, and add them together.

:smack: Ack! Yes, that’s it. Absolutely right, TaxGuy. It’s the W-2G form that I was trying to refer to. (I’ve had to fill one of them out myself, woo-hoo!) Of course all gambling income is subject to taxation. And you wonder how the IRS builds all those big buildings :smiley: I should add one other thing–it’s usually the euphemistically-named “off-Strip” casinos that feature the $1,199 payoffs. Probably because they don’t want to bother with processing the W-2Gs. Thank you, Tax Superhero! :slight_smile:

Achanar wrote, referring to the sequential royal as a “sucker bet”:

That’s true; although, going for the sequential royal to the exclusion of everything else would still be a sucker bet. Also, Scobelete’s survey of video poker machines found that most machines with a payoff for sequential royal were in the 90-95% payout range. At least in Vegas, better machines can usually be found.

Even the regular Royal Flush is on a progressive…starts at $1000 and I have seen it go as high as $1300 before someone wins it. So this bank of machines (at the Green Valley Ranch Casino if anybody is interested) is a normal progressive, double double machine…it just has the added “come on” of offering $100,000 if you get the Royal in a sequential left to right order.

By the way, two years ago I did actually see a woman who hit a sequential Royal on a quarter machine that paid her $12,500 (instead of $1000) for that extra sequential stroke of luck…that was why I asked the OP.