Probability and the Royal Flush
A royal flush, strictly speaking, is just a glorified straight flush. A royal flush in spades is considered the highest possible hand. The chances of achieving it are only 1 in the 2,598,960 possible hands. A great book to teach you how to calculate such numbers for yourself is Probability without Tears by Derek Rowntree. This book would help you with the following calculations:
5/52 X 4/51 X 3/50 X 2/49 X 1/48 = 1/2,598,960
The first fraction is the chance of getting any one of 5 particular cards in the deck (5 in 52, since there are 52 cards in the poker deck). The second fraction is your chance of getting any one of 4 particular cards in the deck on your second card, and so on through the fifth fraction or the fifth card. By multiplying them together you find out what the chances are of coming up with any hand of unique cards. The numerator of our fraction in this case represents our royal flush in spades, while the denominator represents all the combinations possible.
If these odds seem long (many lotteries have better odds), take solace in the fact that your chances of getting one of the four possible Royal Flushes dealt to you are four times as great (1 in 649,740). Now, your chances of actually getting a royal flush in a game of poker depends largely on the game you are playing. In draw poker, if you are dealt four of the cards needed your chance of picking up the final card are actually pretty good, 1/47.