How do you calculate the probability of a hand?

[FirebirdGM]

I was thinking about this today, and realized I have no idea.

Take for example, I want the probability of me getting a royal flush on any given hand. Assuming both of my pocket cards are part of the Royal Flush, what is the chances, and how do you calculate this?

I was thinking something along the lines of this, but I'm probably very, very far off.

Card 1: 20/52 [10-A of any suit]
*
Card 2: 4/52 [One of the others of the drawn suit]
*
Card 3: 3/52
*
Card 4: 2/51
*
Card 5: 1/51

Would that be the odds of getting a royal flush With your pockets and the flop? How would you add the Turn/River? [Assuming I'm anywhere near the right way of calculation]

[galderon]

Card 1: 20/52 [10-A of any suit]
*
Card 2: 4/52 [One of the others of the drawn suit]
*
Card 3: 3/52
*
Card 4: 2/51
*
Card 5: 1/51

Would that be the odds of getting a royal flush With your pockets and the flop?

That's my thinking as well (that is to say, I agree with you, but I may be wrong)...though you need to keep removing cards from the total deck...so it's 20/52 * 4/51 * 3/50 * 2/49 * 1/48...approximately 0.00015%. It's actually happened to me online...play money though.

[galderon]

I think the chance of a royal on the flop or turn is the above chance, times six...meaning that there's one card that you don't care about that occurs in one of the six places:

20/52 * 4/51 * 3/50 * 2/49 * 1/48 * 6 = 0.0009%

The chance that a royal occurs sometime during the community is the above chance of it occuring on the flop, times 21...the two cards you don't care about can be located in 21 different configurations:

20/52 * 4/51 * 3/50 * 2/49 * 1/48 * 21 = 0.0032%

Hmmm...that reasoning makes sense to me, but I can't think of a way to explain it. It might not even be correct!