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How does the PocketFives.com calculator work?

The PocketFives.com Card Calculator employs a mathematical technique known as Bernoulli Trials. Every time you press the calculate button, the calculator runs a sequence of 25,000 independent trials of cards that may land by the end of the hand. The percentages of wins versus losses in this 25,000 trial set are the percentages that appear above the respective seats.

Why Bernoulli Trials?

At a multi-handed table, the possible number of permutations of outcomes per hand can get enormous. As a simple example, consider the classic pocket aces versus pocket kings example in a typical Texas Hold'em game. Because there are five more cards to come in this scenario, and because there are 48 cards left in the deck, there are

48 * 47 * 46 * 45 * 44 = 205,476,480*

possible permutations of hand outcomes. In several of these outcomes, the aces will hold; in a few, the kings will suck out. The situation worsens immensly if you're calculating the chance that pocket jacks will hold up against two random hands preflop. With 9 unknown cards, there are

50 * 49 * 48 * 47 * 46 * 45 * 44 * 43 * 42 = 909,171,781,056,000*

possible hand outcomes! Apply these permutation calculations to random hole cards in a game of Omaha and the numbers become even more staggering.

In order to calculate exact percentages for these examples, your comupter would have to look at each possible hand outcome, determining which hand was better at each step along the way. You would need a lot of computing power and a lot of time to kill for some of these extreme scenarios.

Because the PocketFives.com staff wanted to allow speed and flexibility in the type of game, the number of players, and the randomness or certainty of hole cards, the PocketFives.com Card Calulator has been built using a 25,000 trial Bernoulli set. A 25,000 trial Bernoulli set allows for a golden mean between efficiency and accuracy.

*Note: these values are complete permutations, where the order of the cards that fall does matter. We can reduce this value greatly if we look at unique permutations (i.e., combinations), where a flop of 2 Q J is the same as a flop of J 2 Q . In the aces versus kings example, the number of combinations, dictated by the formula N choose P, is 1,712,304, but in the jacks versus two random hands example, our number of combinations is still greater than 2.5 billion.

How accurate is the calculator?

A Bernoulli trial is an experiment in which a single action, such as flipping a coin, is repeated over and over. The possible results of the action are classified as successes or failures and are dictated by the binomial probability formula,

where P is the probability of k successes in n trials, and p is the probability of success in one trial.

Applying this formula to the card calculator, we can then determine the probability that our result will be within .5% by taking a summation of all probabilities where k successes out of n trials is within the range of p ± .005.

As an example, let's look at the pocket aces vesus pocket kings Texas Hold'em scenario. When the aces and kings are of the same suit respectively, the probabilty (p) that the aces will hold on any one trial is 82.54%. The probability (P') that the calculator (by using 25,000 Bernoulli trials) will be within .5% of this value is

We use a range of k from 20,510 to 20,760 because these values (out of 25,000) correspond to 82.04% and 83.04% respectively. Crunching the equation above yields P' = .9635, so there is a 96.35% chance that the card calculator will produce a value between 82.04% and 83.04%. Using the same formula, the calculator will produce the correct percentage within a .8% variance 99.92% of the time.

It may or may not surprise you that this is a function of the probability (p) of a given hand, not a function of the number of permutations of outcomes. This means no matter how large our sample set gets (as with the example of Jacks versus two random hands above), our error range does not change.