Probabilities of Flush Draws
On a few occasions in the book. I’ve said that you should count a backdoor flush draw as the equivalent of about two outs.
We can use these simple counting methods of computing probabilities to demonstrate why I suggest this.
First, let’s compute the probability of making a two-out draw in the next card.
A draw with two outs would be something like having a 4 
4 
when the flop is 9 8 
3
There are forty-seven unseen cards, two of them are 4s, the probability of making three 4s on the turn is 2÷ 0.47 = 0.04 (4 percent or a one-in-twenty-four chance; note that one-in-twenty would be 0.05, so this is a little more than one-in-twenty).
To make a backdoor flush draw, you need two running cards of your suit. For example, if you hold K Q and the flop is A 7 6 , then you need the turn card and the river card both to be a Heart.
If the turn card is a Heart, then there will be forty-six unseen cards, and nine of them will be Hearts.
Using the multiplication rule, we can calculate this probability as (10/47) x (9/46) =0.04.
The probability of making two running cards to complete a three-flush is the same as the probability of the turn card making you three of a kind when you have a small pocket pair.
In a probabilistic sense, a backdoor flush draw is equivalent to two outs.
Variance
“Variance” is a measure of the dispersion of a probability distribution. The larger the variance, the more likely that a calculation of a mean from a sample result will differ meaningfully from the true mean.
Variance is sometimes thought of as a measure of risk, although, as we’ll see in a moment, it’s not always a good measure of risk in gambling situations.
The computational definition of variance is the mean of the squared deviations from the average. Before we give a formula for it, we need a definition of expected value.
The expected value of X is written E(X) and is equal to
Sum (p(X) x (X)
Where the summation is over call value of X and p (X) is the probability of a particular value of X occurring. Now we can define the variance of X as
V (X) = E (X²)- E (X)²
Let’s look at an example. We can analyze the poker play of a small pocket pair using mean and variance. Generally a small pocket pair will only win if it flops a set.
There are other ways to win, but we’ll ignore them in this model. The probability of flopping a set can be calculated by looking at the probability of not having a matching card on the board.
(48/50) X (47/49) X (46/48) = 0.88
so the probability of flopping a set (or better ) is 0.12 (12 percent, or about a one-in –eight chace).Generally if you flop a set, you’ll win about 88 percent of the time.
That’s based on empirical results from the Turbo Texas Holdem simulations. Now the question becomes how large the pots are.
We’ll analyze the question of whether you should call a raise pre-flop with a small or midsized pocket pair when there are four other active players.
We’ll assume that the poker game is a 10/20 limit and that the pot gets to $300 by the river.
Mathematics of Poker / Other Books on Holdem / Poker and the Internet
