sockobuw
sockobuw
Joined: Nov 27, 2019
  • Threads: 1
  • Posts: 3
November 27th, 2019 at 9:04:20 PM permalink
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.
prozema
prozema
Joined: Oct 24, 2016
  • Threads: 23
  • Posts: 1171
November 27th, 2019 at 9:29:29 PM permalink
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.
"A little luck never hurt any fisherman, that's all I know." - Sig Hansen
DogHand
DogHand
Joined: Sep 24, 2011
  • Threads: 1
  • Posts: 269
November 27th, 2019 at 9:48:29 PM permalink
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
prozema
prozema
Joined: Oct 24, 2016
  • Threads: 23
  • Posts: 1171
November 27th, 2019 at 10:10:53 PM permalink
Looks like we can close the book on this one.
"A little luck never hurt any fisherman, that's all I know." - Sig Hansen
drrock
drrock
Joined: Mar 6, 2012
  • Threads: 1
  • Posts: 42
Thanks for this post from:
CrystalMathtringlomane7crapssockobuw
November 27th, 2019 at 11:32:59 PM permalink
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!
RS
RS
Joined: Feb 11, 2014
  • Threads: 62
  • Posts: 8614
November 28th, 2019 at 2:28:30 AM permalink
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. Iím going with Dr. Rockís answers.
01000101 01110000 01110011 01110100 01100101 01101001 01101110 00100000 01100100 01101001 01100100 01101110 00100111 01110100 00100000 01101011 01101001 01101100 01101100 00100000 01101000 01101001 01101101 01110011 01100101 01101100 01100110 00101110
tringlomane
tringlomane
Joined: Aug 25, 2012
  • Threads: 8
  • Posts: 6033
November 28th, 2019 at 5:32:32 AM permalink
drrock is correct.

For the longest time I did this incorrectly because I didn't take KQJT9 suited out.
7craps
7craps
Joined: Jan 23, 2010
  • Threads: 18
  • Posts: 1933
November 28th, 2019 at 8:48:24 AM permalink
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...
winsome johnny (not Win some johnny)
billryan
billryan
Joined: Nov 2, 2009
  • Threads: 157
  • Posts: 7939
Thanks for this post from:
7craps
November 28th, 2019 at 8:58:26 AM permalink
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.
prozema
prozema
Joined: Oct 24, 2016
  • Threads: 23
  • Posts: 1171
November 28th, 2019 at 9:03:55 AM permalink
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.
"A little luck never hurt any fisherman, that's all I know." - Sig Hansen

  • Jump to: