Dealt 4 to the royal

I'm trying to evaluate a multi-line play I did this week and can't seem to find the frequency that you are dealt 4 to the royal. Any guidance here would be appreciated.

Let's deal the cards one at a time.

5 royal cards x 4 Suits = 20 royal cards in the deck.
You need 1 of them from the first 52.

20/52.

depending on what suit you got, there are 4 royal cards left of the of the remaining 51 cards in the deck.

4/51.

Of the remaining 50 cards, you need one of the other 3 royal cards.

3/50.

49 cards left, two left to make the royal.

2/49.

Now it get's weird. You need to miss the last card.
There is 1 hit and and 47 misses left in the deck.

47/48.

Multiply...

(20/52)(4/51)(3/50)(2/49)(47/48) = probability
1/probability = frequency

I'm coming up with 13,824.26

Somebody double check me please. I'm drinking beer.

Quote: sockobuw

I'm trying to evaluate a multi-line play I did this week and can't seem to find the frequency that you are dealt 4 to the royal. Any guidance here would be appreciated.

sockobuw,

The probability of a dealt Royal Flush is this:

(20/52)*(4/51)*(3/50)*(2/49)*(1/48) = 4*5!*47!/52! = 1.539077E-6, or 1 in 649740 deals.

So, the probability of being dealt 4 to a Royal Flush is this:

(20/52)*(4/51)*(3/50)*(2/49)*(47/48) = 4*5!*47!*47/52! = 0.0000723366, or 1 in 13824.255... deals.

In other words, being dealt 4 to a RF is exactly 47 times more likely than being dealt a RF.

Hope this helps!

Dog Hand

Looks like we can close the book on this one.

You guys are off by a factor of 5. Each of you assumed that the last card drawn must be the non-royal card. In fact, the first card could have been the non-royal card, or it could have been the 2nd card or it could have been the third or 4th. So, you could add 4 more terms to your calculations that multiply to the same fraction as your first term and this would get to the correct answer.

Rather than worry about the order, we can just look at combinations. There are 940 combinations that include exactly 4 royal cards. You have 20 sets of 4 RF holds, AKQJ, AKQT, AKJT, AQJT, or KQJT of each of the 4 suits. And each of these can be matched with one of the 47 cards that do not complete that particular royal.

20 x 47 = 940.

940 / 2598960 = 0.00036168 or 1 in 2764.85106 hands.

So four to the royal is dealt 235 (or 47 x 5) times more often than a dealt royal.

If you look at software like Wolf VP or Video Poker for Winners that shows strategy, you will generally find only 936 occurrences when 4 to the Royal is held in non-wild games. That is because KQJT9 of each suit makes a straight flush, which is generally worth more than holding just KQJT (after discarding the suited 9). When KQJT9 is worth less than 4 to a royal like it is in Deuces Wild, this would not be the case. And, on the other side, 4 to the Royal with a Deuce being dealt would reduce the number of times holding only 4 to the royal since a Royal Flush with Deuces is often worth more.

So, depending on what the original poster wants to do with the information, if we subtract out the 4 instances of KQJT9, the numbers of 4 to the Royal held in most non-wild games would be

936 / 2598960 = 0.00036014 or 1 in 2776.66667 hands.

Have a Happy Thanksgiving! And hope you get dealt 4 to the Royal a little more than expected!

I was gonna say getting dealt 4 to the royal cycle being 13k seems way too damn high. And no way is a dealt royal 47x harder than 4 to the royal. I’m going with Dr. Rock’s answers.

drrock is correct.

For the longest time I did this incorrectly because I didn't take KQJT9 suited out.

Quote: drrock

You guys are off by a factor of 5. Each of you assumed that the last card drawn must be the non-royal card.

agree
in other words, their answers were assuming a specific order of the deal.
the order here (in video poker Q) does not matter at all.

ExceL: ((combin(4,1)*combin(5,4)*combin(47,1))-4)/combin(52,5) =
936/2598960, about 1 in 2776.666667
in words
(4Suits choose 1 * 5RoyalCards choose 4 * 47 other cards choose 1)-4 / 52total cards choose 5

the -4 is for KQJT9 SF as others pointed out but that still includes 4 to the Royal
not subtracting those one should get
940/2598960, about 1 in 2764.851064

I only play about 500 VP hands per week (includes free play) and get 4ttR, on average, 3 times a week
50% of the time 3 times each session played. Many others complain they get the same hand over and over and never hit the Royal.
I never hit the Royal that way in last 3 years of play here in Nevada. NEVER.
go figure...

When I played exactly 1,000 hands per session, I'd get four to a royal about once every other session so less than 1 in 2,000. I'd call it around
once in about 1500.
Strangely, most of my Royals have come from holding three, not four cards, but the number of three hold hands occurs far more often.

Quote: 7craps

agree
in other words, their answers were assuming a specific order of the deal.
the order here (in video poker Q) does not matter at all.

ExceL: ((combin(4,1)*combin(5,4)*combin(47,1))-4)/combin(52,5) =
936/2598960, about 1 in 2776.666667
in words
(4Suits choose 1 * 5RoyalCards choose 4 * 47 other cards choose 1)-4 / 52total cards choose 5

the -4 is for KQJT9 SF as others pointed out but that still includes 4 to the Royal
not subtracting those one should get
940/2598960, about 1 in 2764.851064

I only play about 500 VP hands per week (includes free play) and get 4ttR, on average, 3 times a week
50% of the time 3 times each session played. Many others complain they get the same hand over and over and never hit the Royal.
I never hit the Royal that way in last 3 years of play here in Nevada. NEVER.
go figure...

Very helpful. Thanks for helping me figure out the error I made.

Quote: billryan

Strangely, most of my Royals have come from holding three, not four cards, but the number of three hold hands occurs far more often.

as it should.
my last 10 Royals were:
holding 0: 1 (deuces wild, threw all cards away and it came in diamonds)
holding 1: 0
holding 2: 1 (deuces wild progressive at Plaza)
holding 3: 8 (all hearts or clubs)
holding 4 : 0
holding 5 : 0 (has never happened and I am 63, been playing since 21)

I take a pic of every 4oak or higher and 4ttR
I used to get the 5th Royal card, but no more (in last 3 years)
I get a straight sometimes, no flushes and no Royals from the 4ttR deal

Quote: drrock

You guys are off by a factor of 5. <snip>
940 / 2598960 = 0.00036168 or 1 in 2764.85106 hands.

So four to the royal is dealt 235 (or 47 x 5) times more often than a dealt royal.
<snip>

drrock,

You are correct... That's what I get for posting before coffee :-(

Dog Hand

Hey Dog, no problem. We got to the right answer eventually. That's what these blogs are for. I'm sort of interested in what the original poster wanted the information for. I don't think we have heard back on that.

And you are lucky if coffee is all it takes! ;)

I was playing 0.25 15 play jacks or better and running terribly, losing at a rate of about 3.5%. I was over 4,000 hands dealt into the play (60,000 total hands played) and had not had a dealt 4oak or 4 to the royal. The analysis was to help assure myself I was just running cold and not playing badly. 4 to the royal didn't really mean much to the analysis, but just wanted to know how badly it was going compared to expectation to have 1/47 cracks at the royal. The last 40,000 hands (2,667 deals) it came back around and I ran extremely hot. I was dealt 4oak twice, and 4 to the royal 4 times. I converted on 3 of the 4. It did wonders for my psyche as I was playing a promotion for status, but it was supposed to be just under break even. This was my first time playing that many hands and I had underestimated the variance a bit. even with 15 runouts you can't make good hands out of dogshit.

Quote: sockobuw

I was playing 0.25 15 play jacks or better and running terribly, losing at a rate of about 3.5%. I was over 4,000 hands dealt into the play (60,000 total hands played) and had not had a dealt 4oak or 4 to the royal. The analysis was to help assure myself I was just running cold and not playing badly. 4 to the royal didn't really mean much to the analysis, but just wanted to know how badly it was going compared to expectation to have 1/47 cracks at the royal. The last 40,000 hands (2,667 deals) it came back around and I ran extremely hot. I was dealt 4oak twice, and 4 to the royal 4 times. I converted on 3 of the 4. It did wonders for my psyche as I was playing a promotion for status, but it was supposed to be just under break even. This was my first time playing that many hands and I had underestimated the variance a bit. even with 15 runouts you can't make good hands out of dogshit.

Going 4000 deals without any four to a royal deal should be about 23.67%.

Quote: sockobuw

I was playing 0.25 15 play jacks or better and running terribly, losing at a rate of about 3.5%. I was over 4,000 hands dealt into the play (60,000 total hands played) and had not had a dealt 4oak or 4 to the royal. The analysis was to help assure myself I was just running cold and not playing badly.

How can you blame the deal on playing badly? Just curious.

reread the statement.