Probability and the Straight Flush
The definition of the straight flush includes the royal flush. It basically is made up of five cards all of the same suit and in exact sequence. Rummy players commonly put such runs together in threes and fours after many draws from the pack. However, in poker there are not so many opportunities to make up a hand. Thus the odds of getting such a hand during a poker game are fairly remote. In looking at the royal-flush we found out that the chance of coming up with any one exact hand, say a royal flush in spades, is 1 in 2,598,960.
There are 40 possible straight flushes made up from the standard deck. This is easily illustrated by showing the possibilities of getting a straight flush in any particular suit. Here are the possible combinations:
A K Q J 10
K Q J 10 9
Q J 10 9 8
J 10 9 8 7
10 9 8 7 6
9 8 7 6 5
8 7 6 5 4
7 6 5 4 3
6 5 4 3 2
5 4 3 2 A
Thus all the combinations in one suit comes to ten. If we multiply this by the four suits we can see that there are 40 out of 2,598,960 chances of being dealt a straight flush. Now divide both aspects of our ratio by 40 to get 1/64,974. These are still long odds but better than those of just getting a royal flush. Remember, too, that if you are dealt four of the cards you need for a straight flush, the odds on getting the other card you need are pretty good, 2 in 47.
"Why 2 in 47?" you may well ask. It is because often when you get four cards in a straight flush, all you need to fill it is one card on either end of the straight. For example, you are dealt the 5,6,7,8 of clubs. If you get either the 4 of clubs or the 9 of clubs you have filled your straight flush. The nice thing about going for the straight flush is that even if you fail in making the "flush" portion of your hand you still have a shot at making the straight portion for which you only need a 4 or 9 in any suit. This raises your chances considerably to 8 in 47.
You may have heard the expression, "He was foolish enough to draw to an inside straight". This expression comes from the fact that if you get the outside cards of a straight flush (5,6,8,9 of clubs for example), the chances of getting the 7 are only 1 in 47 half of those in drawing to an "outside" straight flush and only one-eighth of the chance of drawing to a simple straight.