My wife got a royal flush on a video poker machine on two successive pulls yesterday. What are the odds?
From Wikipedia, the odds of getting a royal flush is 649,739 : 1
The odds of getting two of them in a row would be 649,739 x 649,739 :1 or 4.22 x10^11:1 or 422 trillion to one.
Further proof that video poker is rigged to high hell.
Those odds don’t count for one round of discarding. So the odds are more complicated to figure out.
I don’t gamble much, but what does one normally win for a royal flush? Doesn’t that usually trigger some sort of jackpot, even if you’re playing single bet penny slots?
It seems that, according to this, playing Jacks or better on a full-pay machine, the probability of getting a royal flush using simple or optimal strategy is 0.000025, or 1 in 40,000. So two in a row would be 1 in 1.6 billion.
That is not correct. Those only rerlect the odds of getting a royal on your first five cards. The ability to discard and draw makes it a lot more likely.
According to the usually reliable Wizard Of Odds, if you use optimal strategy on a Jacks Or Better machine, the chance of a Royal is 0.000025, or 40,000 to 1.
The OP said two pulls - that implies two deals to me.
No. Here’sa typical payout. You only get the jackpot if you are betting five coins each time. And even then it’s only 4000 times the coin value. So, if you’re playing penny slots you’d win $40.
Yeah, I actually noticed it after I pulled out the probability table.
The 40,000 to 1 is what we hear on the streets around here in small town Nevada. I was curious as to what the internet said. In both cases she says she got A-J on the deal and drew the 10. She dropped $60 winning $800 total then lost two more bucks and quit.
She was betting five coins wasn’t she? Please tell me she was.
Well I hope the payoff was commensurate. 
Speaking as a software engineer for a slot machine manufacturer, this is closest to correct.
Once having gotten one royal flush, the chance of a second on the subsequent game is about 1 in 40,000 - depending on pay style. I’ve always used a scratch value of 1 in 50,000 myself - but perfect play is different with different machines.
There was an interesting story of a fellow who’d won a royal flush in a video poker machine at a grocery store here in Nevada - it seems it was the store’s policy to “clear” the machine after paying it off by giving the customer a free play. And on this occasion, the free play also hit a royal flush, and the customer had to fight the store for it 
Which would make that the worst way EVER to win only $80.
Reminds me of playing craps in Vegas. The tables generally have a “fire bet” present that hits if all the points are made before a 7 goes up. $1 usually wins you $5000 or so. It’s a sucker bet, but it’s a powerful sucker bet-- I must play it every time I play craps for one simple reason: if I’m playing and a fire bet hits, and I didn’t play it, then I’ll never forgive myself for being stingy with $1.
Effective psychology (for the casinos, obviously).
This is true if you play only twice. But your wife evidently had a few hands before and after her two winning ones, so the odds are better than for only two hands.
Can’t a casino set those odds however they want, like a slot machine?
Or do they have to actually replicate a fully shuffled deck of cards?
If the machine plays a card game (eg video poker or video blackjack) then the game must accurately simulate the shuffling and dealing of the cards, per the Nevada Gaming Commission. The machine cannot override that legally:
From the first place that I found with a quick google search (http://casinogardnerville.com/) – be warned, a real annoying popup ad will appear.
AFAIK the machine has to accurately and faithfully reproduce the shuffle and deal of the cards that are played. It is legal to do either of the following:
- shuffle all 52 cards, deal the first five, then deal any replacement cards.
- shuffle all 52 cards, deal the first five, then continuously shuffle the remaining 47 cards until any replacement cards are dealt.
(note that mathematically, these are equivalently random)
More from The Wizard of Odds
I’ve been looking through Nevada Gaming Commission/Control Board PDFs for the exact regulations, and I’m learning a lot as I go If I find the actual regulation I’ll post it; failing that, someone can give them a call.
In the video poker machines I’ve developed, a fully shuffled virtual deck of cards was the preferred way.
The payback is changed and set by adjusting the payouts for various combinations.
The wikipedia link from Peter Morris’s earlier post is of a “6/9” machine, called that because the pays for flush and full house are 6 and 9 respectively. Playing a 6/9 machine is desirable; these are also the most common numbers for a casino to adjust to their advantage, often to 5/8 or even 4/7.
Other video poker offerings, such as a “deuces wild” or “aces bonus” machine (many variants) have very different paytables.
On preview, Scruff has it almost right as well - the first shuffle scenario however is not encouraged.
As alluded to by several posters the original question is not well defined. Assuming 40,000:1 for each throw then the chance of it happening in two throws is about 1.6 billion. However most people play the slots for say at least 100 throws. So during a session the odds of hitting will be approximately 100 X that or about 1 in 16 million. It probably happens to many people a year.