A non political, bi-partisan gaming question.

What is the frequency (I'm looking for the average number of hands dealt) of being dealt:

Four cards to a Royal Flush in Video Poker?

Three cards to a Royal Flush in Video Poker?

Thanks

I would say once every election cycle.....

But Wizard of Odds says the following:

Dear Wizard, In video poker, what are the odds of drawing the following cards to a royal flush?

1. one card
2. two cards
3. three card
4. four cards
5. Dealt a royal flush

I am asking the question because I recently hit a royal flush after holding 2 cards the ace and jack of diamonds and then drew the ten, queen and king of diamonds. I know the odds of drawing 3 cards to the royal must be very high. Then last week , I was sitting next to a man who held the Ace of diamonds and drew 4 cards to complete his royal. I was amazed. Thanks for your answer.

Paul


1. 1/47
2. 1/combin(47,2) = 1/1081
3. 1/combin(47,3) = 1/16215
4. 1/combin(47,4) = 1/178365
5. 4/combin(52,5) = 1/2598960

https://wizardofodds.com/ask-the-wizard/video-poker/probability/

I'm sure that there is faster way of doing this but here is the slowpoke solution (which I prefer):

Each RF4 can have 47 other cards with it (not 48 because then you'd have a dealt royal).

There are 5 different RF4s in each suit and 20 different RF4s in all suits.

So 20 times 47 equals 940 different RF4s can appear in dealt hands.

The total number of dealt hands is 2,598,960. Dividing that number by 940 gives you approximately 2,765.

So RF4 comes up about once every 2,800 hands.

It depends on the pay table and NO wilds,too.
A DEALT RF is rare,but it does happen.
https://wizardofodds.com/games/video-poker/strategy/ultimate-x/jacks-or-better/
You need Javascript to be enabled....

https://wizardofvegas.com/forum/gambling/video-poker/18531-a-little-card-math-help-please/

4OAK: 1:423 hands.
Over the time it takes to play 1,000 hands,you should get 2 4OAK's on average,+ or (-) 100%.
Odds and math people,it's your turn.

My understanding of the original question is what are the chances of being dealt four to the royal, three to the royal, etc. in the first place. Since there are about 2.6 million possible draws we need to divide that by the total number of four card royals, three card royals, etc.

I don't have the will to do that, but it is a different problem than calculating the odds of converting four card royals, three card royals, etc.

Edit: I somehow overlooked Turtle 2's post when I posted this. Everything I lacked the patience to work out is contained in that post.