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:sockobuwI'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

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!

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

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

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...

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:7crapsagree

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.