Hi, bangman121,
Welcome to PokerStrategy.com!
Interesting question.
What poker game is under consideration?
If it is 5-Card Draw for example, the odds of being dealt a pat straight flush are easy to calculate.
There are 2598960 5-card combinations in a 52-card deck.
40 of them are straight flushes -- 4 of which are Royal flushes.
You will always be dealt one of the combinations, so the chances of being dealt a pat straight flush playing 4-card draw are 40/2598960 or 0.001539077%
This is equivalent to 1:64973
No matter how many players there are, you will always get 1 combination, so the odds are always exactly the same.
If the game is Hold'em (whether limit, pot limit or no limit is irrelevant) then the situation is different.
Before the cards are dealt, the odds of flopping a Royal are exactly the same, since your two hole cards, plus the 3 cards of the flop constitute exactly 1 5-card combination.
Once the cards are dealt, though, it is an entirely different story.
If you do not hold two suited cards T to A then your odds of hitting a royal on the flop are 0.
If you
DO hold such cards, then there is 1 3-card combination that will give it to you.
When calculating poker probabilities, you disregard the fact that other players hold cards.
They do, but they are included in the "unknown" cards, so it does not matter which cards they hold.
In hold'em you know two cards, so 50 are unknown, and there are 19,600 ways to group 3 cards from a pool of 50.
So the odds of flopping a royal are 1:19,599
If we hit one or two of our needed cards on the flop, we still have the turn and river to give us the needed cards.
These needed cards are called "outs".
[1]
You can find the odds of hitting any number of outs in the strategy article "
Odds and Outs". If on the flop you need 1 card to complete your straight flush, then it is 22.5:1 against you getting it. If you need two, then it is 1176:1
If on the turn you need 1 then it is 46:1 to get it on the river.
And this does NOT depend on how many other players are in the hand.
The odds of getting any card (or combinations of cards) are defined as: Chances/unknowns
Cheers,
VS
[1] You probably already knew most of this. If you didn't and if I've used any terms you don't understand, then check
the glossary.