Royal Flush question for the Wizard

In video poker there are 6 ways to get a RF (dealt, draw 1, draw 2, draw 3, draw 4 and draw 5). The odds of getting a dealt RF are about 1 in 649,740. How frequently do the other 5 RFs occur?

For example, in 9/6 JOB, a RF is expected to occur about once every 40,390 hands. Assuming perfect play and an infinite amount of playing time, what percentage of RFs will come from drawing 1, drawing 2 etc?

from Ask the Wizard:
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

Quote: bw

from Ask the Wizard:
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

These are the odds given that you are holding 4, 3, 2, 1, and 0 cards, respectively. The last one is incorrect, though, as it should be 4/combin(47,5) or 1/383485.

The probability of a dealt royal is 4/combin(52,5) = 1/649740.

None of these numbers takes into account the probability that you are holding this many cards to a royal.

Quote: bw

from Ask the Wizard:
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

Sorry, but I did not make my question clear. I know that there is a 1 in 47 chance of drawing a RF from RF4 and a 1 in 1,081 chance of drawing a RF when holding RF3. However, my question has to do with the frequency of occurrence among the ways in which a RF occurs, i.e drew 1, drew 2, drew 3 etc.

My limited experience is that RFs occur more frequently when drawing 2 cards than when drawing 1 card (presumably because RF3 is held far more frequently than RF4).

So given a sufficiently large number of RFs, what percentage will come from drawing 1 card, 2 cards, 3 cards etc?

For 9/6 Jacks or Better, the odds are:

dealt (hold 5 cards): 1 in 649,740
hold 4 cards: 1 in 130,503.3333
hold 3 cards: 1 in 102,192.4836
hold 2 cards: 1 in 205,912.9112
hold 1 card: 1 in 1,186,106.165
hold 0 cards: 1 in 11,814,385.09

This takes into account perfect strategy, so on many hands, you will not be able to get a royal. the reason why the odds of a royal flush are so low with holding 0 cards is because it is quite rare to discard all cards in jacks or better.

As a percentage of all royal flushes:

dealt: 6.21%
hold 4: 30.93%
hold 3: 39.50%
hold 2: 19.61%
hold 1: 3.40%
hold 0: 0.34%

Wow! I am very impressed with your answer and with the speed of your reply. One final question: are the percentages significantly different for perfect play at other vp games, e.g. 10/7 DB, 16/10 NSUD, 10/6 DDB?

Thanks.

10/7 DB

7.39%
36.81%
36.89%
15.10%
3.64%
0.17%

10/6 DDB
6.27%
31.24%
38.27%
18.94%
5.06%
0.22%

16/10 NSUD
6.63%
30.32%
51.59%
9.52%
0.00% - notice here that you will never hold one high card in deuces, so drawing 4 to a natural royal is impossible
1.95%