### Features

#### Simulate Button

The simulate button allows you to perform a hot-and-cold equity simulation between two or more ranges of hands. The simulator calculates what happens on average when all of the players go all-in. For example, here are the results of a race between AsAh and JdTc:

Let's take each piece of data in turn:
• 1,712,304 trials - the number of all-in matchups that were simulated.
• (Exhaustive) - in this case, all possible hand and board combinations were generated. (More details on this below.)
• Hand - this column should be self-explanatory.
• Equity - this is the average percentage of the pot won, after taking ties into account.
• Wins, Ties - the number of times each player won or tied for each hand dealt.
Here's another example racing a pair of tens against three random hands with a low flop:

There are two possible algorithms employed, depending on the game selected and the number of hands possible for each player.

• Exhaustive (fully-enumerated) results - All possible matchups between all players are generated, and equity statistics are compiled. This yields an exact answer, and is not subject to random variation.
• Randomized (monte-carlo) results - A certain number of random matchups are generated (see Random Range Generation ), and for each matchup, hands and/or boards are completed and a single statistical datapoint is collected. For most simulations, 600,000 trials are conducted. As the results are random, results may vary slightly from run to run. In practice, the variation is rarely more than a few tenths of a percent.

#### Unrolled Simulation

An 'unrolled' simulation is a convenient way to display a simulation between a number of hands for all streets. The following unrolled simulation charts the changing fortunes of two hold'em hands from before the flop to the final river card:

#### Count Button

The count button allows you to count the number of hands in a given range or ranges. Counting is supported for hold'em, omaha, omaha-8, stud, and stud-8. Here's an example count for omaha:

Let's take the columns from right to left, focusing on the first row:
• The Base Count is the number of hands possible for a range in a vacuum. In this case, we see that there are 6961 ways to deal four cards where at least two kings are present, comprising 2.57% of all 270725 possible omaha hands.
• The Optimized Count removes hands that can't exist in our range due to other hands present and any dead cards. In this case, the king of spades is already accounted for in another hand, bringing our count down to 903 hands, or 0.33% of all possible omaha hands.
• The Hand column is the, um, hand.

#### Graph Buttons

##### HvR (Hand vs. Range) Graphs

The hand vs. range graph button allows you to view a graphical representation of the possibile equities of a hand on the next round of betting when the other hands are unknown (that is, the other hands are still ranges). For instance, here is the graph of a pair of sixes vs. a hand in the top 15%:

The HvR graph shows as that the sixes have very high minimum equity on about 12% of flops - this is when the sixes flop a set. Moving over, we see they have at least 50% equity about 30% of the time.

##### HvH (Hand vs. Hand) Graphs

The hand vs hand graph button allows you to view a graphical representation of the possible equities of a hand on the next round of betting when all hands are turned face up. Let's take a look at the same example we used for HvR - a pair of sixes vs. a hand in the top 15%:

The HvH graph shows us that the sixes have over 75% equity on about 20% of flops hand vs. hand. These represent the times that they flop a set or a monster draw. Moving a little further to the right on the graph, we see that the sixes have more than 50% equity on about half of all flops. Finally, looking at the far right hand of the graph, we can see the sixes can't do better than about 15% equity on about 40% of flops. This reflects those times that the sixes are beaten by a bigger set or pair.

##### Raw Graph Data

If you graph a simulation of AA vs. T9s and hit the 'Raw Data' link, you might see something like the following:

This is a raw dump of the data used to power the graph in json format. Each hand is listed followed by a set of equity bucket counts (more on this below) between square brackets. For instance, "AA",["144","3516","1296",.... a bunch more numbers here....]. Each of the numbers represents the number of times the given hand received a certain amount of equity. For example, if there were ten buckets listed like so: "AA", ["1", "3", "4", "5", "5", "7", "7", "8","9","9"], that would mean that aces had 0-10% equity once, 10-20% equity three times, 20-30% equity four times, ..... and 90-100% equity nine times.

#### To English Button

The "To English" button allows you to translate a range into English. For example:

#### Rank Button

The rank button allows you to rank hold'em, omaha, and omaha-8 hands. Ranks range from 1 (highest) to 100 (lowest). For example:

We can see the average, best and worst rank for each of our ranges of hands for each of the available rankings.

For KK23, the best possible 10-handed rank is 4 (for the double suited version), and the worst rank is 15 (for the rainbow version). Our average comes out at 9.13. This average is computed by generating all possible KK23 hands and tabulating the ranks of each. As a result, some combinations (such as the three suit combinations which are more common) will be weighted more heavily than others (such as the 12 rainbow possibilities).

For AdJd3s4s, only one hand is possible, so the average, best, and worst ranks are all the same.

#### Random Range Generation

All of the games currently supported by propokertools other than razz use the same general strategy for generating random ranges of hands when performing monte-carlo simulations.

• Generate a list of all possible hands for each player given their range.
• Pick a hand at random from each player's list.
• If any card appears in two places, our random choice is invalid - go back one step.
• At this point, we have a fair, random tuple of hands and can proceed.

#### Hand Rankings

The omaha, omaha hi/lo, hold'em, five-card omaha, and five-card omaha hi/lo simulators include support for the top N% of hands, where N is a number between 1 and 100. These hand orderings were generated by a computer program. For details on the algorithm employed, read this blog post .