In Part 1 of this lesson we discussed the number of possible two-card combinations that could be dealt before the flop, and arrived at the figure 1,326.
The arithmetic for solving combinations involves setting up division equations. First, we determine the top number in our division equation (the numerator, if you've forgotten) by multiplying components of our mathematical universe (i.e., the 52 card deck), in descending order. How many components? Select as many as there are choices. In this case, we're trying to determine the possible number of two-card combinations, so we multiply 52 x 51 (which equals 2,652) to get our numerator. That's step one. To determine the bottom number in our division equation (the denominator) we multiply each component of our choices in ascending order, in this case: 1 x 2. That's step two.
Step three is to set up a division problem. We divide 2652 by 2 for the answer:1,326.
How many different flops are possible? What you're trying to determine here is how many possible choices of three cards can be taken from a universe of 50 cards. Why 50, instead of 52? The two cards in your hand are not included in this universe because they cannot possibly appear simultaneously in your hand and on the flop. Setting up the problem involves dividing the product of (50 x 49 x 48) by (1 x 2 x 3), or 117,600 by 6. It's that easy. There are 19,600 possible flops - and it's a handy number to know. Let's use it to solve a practical Texas Hold'em problem - one we're certain you'll encounter the very next time you play.
If you have a pair in the hole, what are your chances of flopping a set - or better? It's actually easier to calculate the number of ways to miss making a set. Let's try that. If, for example, you hold 8♦8♣ in your hand, there are two other cards in the deck that will form at least a set from a universe of 50 unknown cards - excluding the rather remote possibility that the flop itself is a set.
So for all practical purposes, if the 8♥ and 8♠ are the only two cards that will make at least a set; the remaining 48 cards will miss. If the first card up on the flop is neither the 8♥ nor the 8♠, then 47 of the remaining 49 cards will also miss, and if the second card brings no help, that third and final flop card will not be the 8♥ or 8♠ 46 times out of the 48 remaining unknown cards.
Your next step is simple. Just multiply the fractions, as follows (48/50 x 47/49 x 46/48). When you multiply the numerators (top numbers) you get 103,776, and when the denominators are multiplied the answer is 117,600. This means you won't flop a set or better 103,776 out of 117,600 times. By subtracting 103,776 misses from the universe of 117,600, you are left with 13,824 hits. Now you know that flopping a set or better figures to occur 13,824 times out of 117,600.
To reduce it, divide the numerator and denominator by 13,824. When you do, you'll find yourself left with 1/8.5. If you divide 1 by 8 (or 13,824 by 117,600), you get 0.118. Expressed as a percentage, you'll flop a set or better 11.8 percent of the time that you hold a pair in your hand.
If you figure to flop a set 11.8% of the time you're holding a pair in your hand, how do you express that in odds? There is a relationship between percentages and odds, and understanding this relationship will help whether you're betting on horses, playing cards, or merely wondering what the chances are that it's going to rain. To change a percentage to odds, subtract the percentage from 100; then divide the result by that same percentage. If something has a 20 percent chance of occurring, this is the calculation: (100 - 20) / 20, or 80/20, which is equal to 4. Thus the odds against something with a twenty percent chance of occurring are said to be 4-to-1.
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