Overview
The basis of all gambling theory is the concept of “Expected
Return” or “Equity”. Simply stated, your
equity for a gambling event is the average amount you will get back in return,
over an infinite set of events, for each unit bet. If your equity for the event is greater than
1.0, you will win in the long run, if it is less than 1.0, you will lose, and
if the equity is exactly 1.0, you will break even. Obviously, in a small finite number of events
you could win with equity less than one and visa versa, but in the long run if
you make bets with an equity less than one you will lose. That’s exactly why the casinos are getting
richer every day.
For example, if the event is a hand of Blackjack there are two opponents, the player and the dealer – each having a known probability of winning the hand (these odds have been calculated and published). To simplify this discussion we will assume that you are betting one dollar each hand and you either win or lose a dollar with a tie being a wash. In the most favorable casinos, the house advantage is about 51%. That means that the house will win 51% of the hands and you will win 49%. If you play 100 hands, you will win 49. For each hand you win you will receive your dollar back plus a dollar from the house – or two dollars. So, if you win 49 hands you will receive $98 for the $100 bet. In that case, the expected return is $0.98 for each dollar bet. When there are only two opponents and a single payout size, this equity is easily calculated since it is double your winning percentage.
If there are additional betting options and variable size payouts (like a bonus for Blackjack, splitting pairs, etc.), the equity can be calculated by multiplying each payout size by the probability of getting that payout and adding them all up. It’s more complicated but yields the same result – your expected return for each dollar bet.
In the more general case where there are multiple opponents, like a hand of Holdem, the equity may be computed as well. Assuming that you are playing in a six handed game and everybody antes $1. The cards are dealt out with no further betting and the best hand wins (ignoring ties for the moment). If you expect to win 20% of the hands (because you are better or luckier than most of your opponents) you equity may be computed as follows for 100 hands:
Equity = 6 * 20 / 100 = 1.20
The above formula represents the amount you receive when you win ($6) times the number of hands you win (20) divided by the total number of hands played (100).
Ties may be factored in as well by counting a tie as a fraction of a win (i.e., a three way tie would count as 1/3 of a win). For example, if you won 18 hands outright and tied 8 with exactly one other person, the equity computation would be:
Equity = 6 * ( 18 * 1.0 + 8 * .5 ) / 100 = 1.32
Hand Values
This concept of equity can be applied to a Holdem hand at any instant of time to determine that hand’s relative value compared to all other possible hands. In this case, the value of the hand is not an indicator of how much actual money you may win or lose since there will be raises and players folding in the future which cannot be predicted. What it does indicate is that from that instant, if everyone made a single bet and there were no raises or drops, what the expected return would be. This concept can be applied at any point in the hand and is the basis for Holdem Coach hand values. Over the course of the hand, as new cards are dealt and players fold, the hand value varies up and down but at any point in the hand the value indicates the strength of the hand at that time.
The initial application of this concept occurs immediately after the first two hole cards are dealt to all of the players but no betting has taken place. Using billions of simulations to determine the winning percentage of each possible starting two cards in a game, the equity may be determined for each size game (2 to 10 players). The best starting hand is a pair of Aces, an average hand is King-Seven unsuited, and the worst hand is Seven-Two unsuited. In a six handed game the equity for Aces is 2.96, for King-Seven is 1.00, and for Seven-Two is .51. These hand values represent those in a game with six players, other game sizes will have slightly different values.
Once the relative equity is known for a hand, heuristics for action can be developed based on the value. A possible set of rules might be: before the flop fold any hand with value less than 1.0; only play hands between 1.0 and 1.1 in a late betting position (e.g., dealer); raise with any hand with a value greater than 2.0, etc. Again, remember that these values are relative not absolute and don’t take into account folds, money in the pot, future raises, etc. These values are most useful prior to and immediately after the flop to give you an idea of the prospects of your hand. In later rounds the pot odds and your read on how your opponents have been betting should weigh heavily in your decisions.
User Interface
Holdem Coach has been developed to be used as an aid for on-line poker players for advice and tracking purposes. As such, the interface is light weight and compact so that it can be run side by side (actually top to bottom is better) with the on-line poker client software screen. The purpose of Holdem Coach is two-fold:
· It allows the entry of the cards as they are dealt and provides the hand value and various simulated and computed statistics for the player’s hand at any instant in time.
· It tracks the results for a session of play and computes averages which allow a player to statistically track his results and compare sessions.
Using the sample screen below, the user interface is explained.
Hole/Common Cards
These represent the cards that have been dealt. To add a card, select it from the ‘Add Card’ area and the card is inserted into the lowest available slot. The buttons for cards already selected will be grayed out. To remove a card that has been selected, simply click on that card.
Simulated Hand Values - Deal
This represents the initial hand value and winning percent for the two hole cards prior to any betting. It is not affected by any cards that are subsequently added and will change automatically if the number of deal players or the hole cards are changed. These values have been predetermined for each game size (i.e., number of players) by running one billion hand simulations for each. The total list of all hand values is at the end of this document.
Simulated Hand Values - Current
This represents the current value and winning percent of the complete hand as shown. If the number of players or the cards in the hand are changed, the values are cleared and will need to be recomputed.
Computed Percentages
This area displays the percentage of actual possible final hands that your hand will either beat, tie, or lose. The ‘All’ line compares your hand with all possible hands. The ‘Avg’ line creates a realistic set of remaining hands and compares you hand with those hands. This gives a better idea of how you will do in an actual game where the weak hands have dropped. The percentages that are entered in the preferences (see below) are used to determine how many hands remain at each stage of the hand (flop, turn, river). All of these averages are actually computed, not simulated, by looking at each possible hole card set and each possible set of cards that may complete the hand. In addition to the percentages, the average value for all hands used for computing the percentages is displayed. This gives an estimate of how strong your competition is likely to be.
Best Hand
If there are at least five contiguous cards (no empty slots created by deleting cards), this is the best poker hand that can be made from the cards currently shown in the hand.
Compute
Clicking this button will calculate the values and percentages for the current hand. This computation can only be done if there are at least three common cards and all cards in the hand are contiguous. If you want to accumulate accurate statistics for the number of players at each stage, then the current players and cards need to be added and Compute clicked – even for hands where you have folded.
Details
Displays the hand details in a separate window (see
description below).
Clear
Clears all cards from the hand and resets the appropriate display values.
Win / Tie / Loss
If a session has been started (see below), these buttons are used to record the result of the current hand. The cards are then cleared in preparation for the next hand.
Start Session / End Session
This button (whose label is toggled between ‘Start’ and ‘End’) initializes and completes a session. When a session is started, the user chooses an existing ‘Session Description’ or types in a new description. All sessions which have the same description will be grouped together on the same results web page with totals and averages (see Home Page below).
Pause Session / Resume Session
This button (whose label is toggled between ‘Pause’ and ‘Resume’) stops and restarts the timer for the session. This is used so that the time-based statistics that use playing time will be correct. Typically a session would be paused when you choose the ‘Sit Out’ option from the Poker software.
Undo
This button rolls back the last Win/Tie/Loss button push and restores the cards, the number of players, etc. to their previous settings. It is used in case a mistake is made – such as the wrong button being pushed, a button pushed accidentally, or incorrect data when the button was pushed.
Stats
Displays the statistics for the current session in a separate window.
Preferences
Displays the ‘Preferences’ window which allows the user to customize Holdem Coach settings. The following preferences are available:
· Setting the number of deals used in the simulations used for the Simulated Hand Values. This is configurable to allow for adjustment based on the processing speed of the computer. This number should be set as high as possible which still gives an acceptable time to process the computations.
· Setting the percentage of players which on the average remain at the flop, turn, and river. Since there is such a wide variation in these percentages depending on the stakes of the game, this allows you to set these numbers to match the games in which you play. The actual numbers for these percentages are tracked as part of the session statistics and after a few sessions, you should have a pretty accurate idea of what the settings should be. These percentages are used to filter out hands when computing the ‘Avg’ Computed Percentages.
· Choosing which data is shown on the statistics web pages and allowing the regeneration of the web pages if this preference is changed.
· Changing the screen configuration from ‘Wide’ to ‘Square’. For high screen resolutions (1024x768/normal fonts and higher), the wide configuration allows most poker site software windows and Holdem Coach to reside on the screen simultaneously with no overlap. (See Adjusting Screen Resolution and Window Positions section below). For lower resolutions there is going to be overlap and the square configuration may be preferable. After changing shape selection, the program has to be restarted for the change to take effect. The wide configuration, which is the default since it is preferable in most cases, is shown below:
Home Page
The Holdem home page provides links to the documentation, to contact customer support, to the JokerPoker Software home page, and the session statistics.
Analyze
Displays the Analyze window which allows analysis of more general situations. For example, what would be the winning percentage of pocket kings vs. pocket aces. When the analyze window is initially displayed, it contains the cards and number of players that are currently in the main window. That way the analyze window can be used during play to obtain a quick value calculation against hypothetical hands. For example, If you want to compute the value of your hand assuming that one of the opponents has pocked aces, you can display the analyze window and enter pocket aces in one of the blank hands.
By default in the Analyze window the next card will be placed in the slot following the card previously placed. To place a card in a specific location, click on a blank location and a Joker will appear. The next card selected will replace the Joker. As for the main window, to delete a previously placed card, click on it.
Exiting Holdem Coach
To exit the program, click on the ‘X’ in the upper right corner of the window.
Hand Details Window
The Hand Details window shown below provides additional information to complement the information in the main window.
The hand and hand statistics are repeated from the main window for convenience. The information in the remaining three sections is as follows:
Current Result
This section displays how your hand compares to all other possible hole card hands as the current time without projecting results by adding additional community cards. For example, after five cards have been dealt your hand is better than all but 50 of the 1081 possible hands. Of the hands that beat you, seven already contain three of a kind. Of the 1031 hands that you beat, 347 of those hands have lower pairs.
Final Hand
Distribution
This section gives the number of possible combinations for the remaining community cards that will result in your final hand being of each type. The raw numbers as well as the percentages derived from those numbers are displayed.
Interpreting Hand Values
Holdem Coach is meant to be used simultaneously with on-line poker clients. It takes some practice, but it is possible to enter the information into Holdem Coach while playing the hands and use the resulting computed values as a guide to making decisions. The tracking capability also allows a player to track improvement (or decline) in his level of play over time by comparing session results.
One of the many excellent books for Holdem poker,” Hold ‘Em Excellence” by Lou Krieger, has a ‘Start Chart’ for recommending what action to take before the flop with each of the 169 possible initial hole card combinations. In addition, this book contains information regarding Power Ratings using a formula developed by Bill Chen which bases actions on the Power Rating of the hand. Mapping their recommendations to the hand values computed by Holdem Coach, a possible set of rules which approximates the Krieger/Chen recommendations would be as follows (where the number is the minimum hand value for calling in each case and the hand position represents betting position where the dealer is the latest position since he bets after everyone else):
|
Hand Value |
Action |
Units to Call |
|
1.00 |
Call in middle or late position only with no previous raises |
1 |
|
1.20 |
Call in all positions with no previous raises |
1 |
|
1.30 |
Call in all positions with one previous raise |
2 |
|
1.50 |
Call in all positions with two previous raises |
3 |
|
2.00 |
Raise in all positions |
|
A somewhat more difficult task is to develop simple heuristics based on hand value for post-flop hands. At this point of the hand the betting pattern for the hand (i.e., aggressive or passive), the amount of money in the pot, the individual opponent’s tendencies, and the number of players that remain need be factored into the decision making process. In this case the Computed Percentages section can be used in conjunction with the hand value to get an overall picture of the strength of the hand. A particularly useful piece of information is the difference between the ‘All’ win percent and the ‘Avg’ win percent. If the ‘Avg’ is considerably lower than ‘All’, this is an indicator that the hand value may not be as good as it would appear, since the hand does not play nearly as well in a realistic game condition as it does when compared to all possible hands.
Until these post-flop heuristics and hints are included in Holdem Coach (see future enhancements below), the main thing to remember is that the hand value represents the relative value of the hand compared to all of the possible hands and that a value of 1.0 is average. Also, remember that the maximum value for a hand is equal to the number of players remaining in the hand. That means that a hand with a value of 2.0 in a three handed game will win 2/3 of the time while the same hand will win only 1/3 of the time in a six handed game (but you will win twice as much). Any time the hand value approaches the number of players (e.g., a value of 3.8 in a four handed game), then that hand is almost a lock to win.
Session Statistics
Holdem Coach has two primary functions – to calculate the value of a hand at any point during the deal and to accumulate statistics for the deals that are played. The values have been discussed previously and the statistics are discussed in this section.
Below is a list of all of the possible data that the user can have as part of the session report. Each of these items may be removed from the reports by making selections in the User Preferences window. Each possible data item is listed below with a brief explanation of what that data item represents and how it is determined and collected. The data for each session is accumulated with all other sessions with the same description and a report is generated showing each session listed as well as the totals for all sessions.
Date – The date when the session started.
Start Time – The time the session started.
End Time – The time the session ended.
Total Time – The total time duration of the session (i.e. end time minus start time).
Playing Time – The total time actually spent playing. This is the total time minus any time that occurred while the session was in ‘Paused’ mode.
Total Pots – The total number of deals played during the session.
Pots Won – The number of pots won – accumulated when the ‘Win’ button is clicked.
Pots Tied – The number of pots tied – accumulated when the ‘Tie’ button is clicked.
Pots Lost – The number of pots lost – accumulated when the ‘Loss’ button is clicked.
Wins Folded – The number of hands that were folded but would have won the pot if had stayed in until the end of the hand – accumulated when the ‘Loss’ button is clicked.
Win Rate – The effective number of pots won adjusting for ties. For example, 6 wins, a 2 way tie, and a three way tie would produce a win rate of 6 + .50 + .33 = 6.83.
Win Percent – The percent of pots won which is computed using the Win Rate divided by the Total Pots.
Fold Percent – The percent of pots folded which would have won which is computed by using the Wins Folded divided by Pots Lost.
Pots / Hour – The number of pots played per hour which is based on Total Pots and Playing Time.
Players / Deal – The average number of players who are present when the hole cards are dealt. This value is derived by totaling the Deal Players and dividing by Total Pots.
Players / Flop – The average number of players who are present when the flop is dealt. This value is derived by using the Current Players value when the Compute button is clicked when exactly three common cards exist. This value is used as the default in the drop down list when the user clicks ‘Win’, ‘Tie’, or ‘Loss’ button.
Flop Percent – The percent of the players present at the deal who are still in the hand at the flop – i.e., the Flop Players divided by the Deal Players.
Turn Percent – The percent of the players present at the deal who are still in the hand at the turn.
River Percent – The percent of the players present at the deal who are still in the hand at the river.
Average Value – The average of all the hole card values for all of the hands dealt.
Percent Playable – The percentage of hole card hands where the value is greater than or equal to 1.0.
Start Stake – The player’s stake at the beginning of the session.
End Stake – The player’s stake at the end of the session.
Won / Lost – The difference between the start stake and the end stake.
Using Holdem Coach
To help with getting started using Holdem Coach, the following list outlines a set of steps and tips that I have used and developed while playing on-line.
1. Get an account of one of the on-line poker sites. I use Pacific Poker because they have one of the cleaner interfaces and have the option of playing for free. Their web site is www.pacificpoker.com. Sign up for a free account and they will credit you with $1000. Play some hands for low stakes (e.g., $1/$2) to get the feel for their interface and how the game flows. Once you are comfortable playing on their site, it’s time to roll in Holdem Coach.
2. Fire up Holdem Coach and start a session. Click ‘Start Session’ and give the session some meaningful description, such as “Pacific Free 1/2”. When you start subsequent sessions, this description will appear in the drop down list so you can accumulate stats from multiple sessions with the same description. You are now ready to play using Holdem Coach as a guide.
3. Set the number of players at the time of the deal. Once the hole cards have been dealt, count the number of hands with cards and enter that value in the Deal/Players dropdown list. I find it is easiest to mentally split the table into two halves with an imaginary vertical line. Each side has five possible hands and it is easy to quickly get a count of each side without actually counting – sort of like visually interpreting a dice roll without actually counting the pips.
4. Set your two hole cards. This is a matter of setting the suits and cards. Once the two hole cards are set, the value for the hand, based on the number of players, is automatically displayed. If you make a mistake either selecting the cards or setting the number of players, correct the mistake and the value will automatically be adjusted.
5. Determine your pre-flop strategy. This action can be based on the table in the previous section or you can devise your own strategy. The table above is fairly conservative when compared to the usual low limit ‘No Foldem Holdem’ play but is very close to being mathematically correct. Particularly while practicing for free, you may want to play a looser game since you get to participate in more hands. In any case, I would recommend that you always fold with a hand value less than 1.0 (unless you are on the blind) since you are already an underdog and making a losing bet.
6. Set the number players at the flop. In order to get an accurate hand value for your hand after the flop, the number of players who remain to see the flop needs to be set in the Current/Players dropdown. If you are interested in the “Players at Flop” statistic which is accumulated for the session, then you will need to enter this number, even when you drop prior to the flop, or correctly enter that number when you click Win/Tie/Loss. If you are not interested in that statistic, then it is not necessary to enter it.
7. Add the flop cards to the community card set. Again, if you have dropped prior to the flop and are not interested in the “Players at Flop” statistic then this is not necessary.
8. Click the Compute button. For the remainder of the hand, unlike the initial two cards, the hand value is computed based on a real-time simulation (the simulations were pre-determined for the values of the hole cards).
9. Determine your post-flop strategy. Devising a set of heuristics based on the hand value after the flop is more complicated the prior to the flop. The hand value will give you a relative worth of the hand compared to all other possible hands – with hands below a value of 1.0 being below average and hands over 2.0 being well above average. The hand value, along with the computed percentages, should give you a pretty accurate view of the strength and prospects for your hand.
10. Add the turn and river cards, set the number of players, and click Compute. If you have dropped from the hand at this point this is not necessary unless you want to accumulate accurate Turn and River Player statistics.
11. Click ‘Won’ or ‘Tie’ or ‘Loss’. A window pops up to gather additional information about the hand. The information gathered depends on the button clicked. The number of players at the flop may be entered at this time if you did not set the flop cards and click ‘Compute’. When you click ‘OK’, the result of the hand is recorded and the cards are cleared in preparation for the next hand.
12. End the session. When you are finished playing for this session, click ‘End Session’ to record the data from the session and display the final statistics. To see the data from all sessions click ‘Home Page’.
That’s pretty much all there is to it. At first it will seem awkward and you will spend most of your mental energy entering data rather than playing poker and watching what is going on at the table. After some practice, you will be entering the data automatically without distracting you from the game itself. I have found that when it is your turn to act and your action is obvious, it is best to take your action prior to entering data. This prevents slowing down the game unnecessarily.
Adjusting Screen Resolution and Window Positions
Holdem Coach plays best with screen settings of 1024x768/Normal Fonts or higher. With that resolution it is possible to position the Poker Client program and Holdem Coach on the screen with no overlap. At a resolution of 1024x768/Normal Fonts the task bar, which is normally along the bottom of the screen, will need to be moved or hidden to allow full screen height. At higher resolutions this is not necessary.
This section describes how to determine and change the current screen settings and how to move the task bar.
Determining and Changing
Screen Resolution
· Go to Start Menu > Control Panel and double click on Display
· Click on the ‘Settings’ tab
· The screen resolution will be represented by a slider bar with the resolution shown below. A 17” monitor or any decent 15” monitor should be readable using 1024x768 resolution with normal size fonts.
· To set the resolution, move the slider bar by placing the mouse cursor over the slider, holding down the left mouse button, and dragging the slider to the desired resolution.
· To set the font size, click the ‘Advanced’ button. On the advanced settings page under DPI Setting, choose normal size. (If you are using a higher resolution, such as 1280x1024, you can set the fonts to large size if the normal fonts are too small)
Moving the Task Bar
· Find a spot on the Task Bar (the task bar is the row of buttons along the bottom of the screen) that is not covered by a button.
· Click the right mouse button to display the task bar menu. If ‘Lock the Task Bar’ has a check mark next to it, click on Lock the Task Bar to remove the check. This allows the Task Bar to be moved.
· From an unused spot on the Task Bar click and hold down the left mouse button.
· Move the mouse up and to the right and the task bar will be displayed along the right edge of the screen – at that point, release the mouse button.
· To return the Task Bar to its original position later, reverse the procedure by dragging down and left.
Positioning the Poker
Client Software and Holdem Coach
· Start the Poker Client and Holdem Coach
· Move the two windows such that the Poker Client is on top and Holdem Coach is directly below it.
· To move a window, place the mouse cursor on the top title bar of that window and hold down the left mouse button.
· Holding the left button down, move the mouse cursor to the desired position and the window will move with the cursor.
· When the window is in the desired position, release the mouse button.
Analyze
A logical extension to Holdem Coach is the capability to analyze more varied starting propositions and give a larger set of results. The Analyze function fits that role. Analyze may be run during play to provide additional calculations (see user interface description above) or can be used in a stand-alone mode independent of a session. This page displays win/tie percentages and the percentage of the time for getting each hand type as shown below.
Future Enhancements
Some of the possible enhancements to Holdem Coach are listed below. Whether these features are added and the order they are added will depend on user feedback.
1. Adding a hint feature which will outline possible strategy based on the hand value and other factors of the hand such as pot size, betting pattern, number of players, etc.
2. Adding the capability of adding cards via the keyboard as an alternative to the mouse.
3. Adding the capability of session management (i.e., deleting or modifying session data)
4.
Enhancing the data provided by the Analyze
window.
Hand Values Prior to Flop
Initial hand values for each game size (i.e., number of
players). Hands with a ‘*’ are suited.
2
3 4 5
6 7 8
9 10
------------------------------------------------------------------
AA
1.71 2.20 2.55
2.79 2.96 3.04 3.10
3.12 3.11
KK
1.65 2.07 2.34
2.49 2.59 2.61
2.65 2.64 2.61
QQ
1.60 1.96 2.15
2.24 2.28 2.27
2.26 2.24 2.23
JJ
1.55 1.84 1.96
2.01 2.01 1.99
1.97 1.95 1.94
TT
1.50 1.73 1.81
1.81 1.80 1.76
1.75 1.72 1.71
99
1.44 1.60 1.64
1.63 1.60 1.56
1.55 1.56 1.54
88
1.39 1.51 1.51
1.48 1.45 1.42
1.41 1.43 1.44
77
1.33 1.39 1.37
1.35 1.31 1.30
1.31 1.33 1.37
66
1.26 1.30 1.26
1.22 1.21 1.22
1.23 1.26 1.30
55
1.20 1.20 1.16
1.11 1.11 1.12
1.15 1.19 1.24
44
1.14 1.11 1.05
1.03 1.03 1.06
1.11 1.16 1.21
33
1.07 1.01 0.96
0.95 0.97 1.02
1.07 1.14 1.21
22
1.01 0.93 0.88
0.89 0.93 1.00
1.05 1.13 1.18
AK
1.31 1.45 1.55
1.62 1.67 1.71
1.74 1.72 1.72
AQ
1.29 1.40 1.48
1.52 1.55 1.57
1.58 1.57 1.55
AJ
1.28 1.37 1.42
1.45 1.46 1.46
1.46 1.45 1.43
AT
1.26 1.33 1.36
1.38 1.38 1.37
1.38 1.35 1.33
A9
1.21 1.25 1.24
1.23 1.22 1.19
1.16 1.14 1.12
A8
1.20 1.21 1.20
1.17 1.16 1.13
1.10 1.08 1.05
A7
1.17 1.17 1.15
1.12 1.09 1.07
1.05 1.02
1.00
A6
1.16 1.14 1.09
1.08 1.05 1.02
1.00 0.98 0.96
A5
1.15 1.14 1.12
1.10 1.08 1.06
1.05 1.04 1.02
A4
1.14 1.11 1.09
1.06 1.05 1.04
1.02 1.00 0.99
A3
1.12 1.08 1.05
1.03 1.02 1.00
1.00 0.97 0.96
A2
1.11 1.05 1.01
0.99 0.98 0.96
0.95 0.94 0.92
KQ
1.23 1.33 1.41
1.46 1.50 1.52
1.53 1.52 1.51
KJ
1.21 1.30 1.35
1.38 1.41 1.42
1.41 1.40 1.39
KT
1.19 1.26 1.30
1.32 1.34 1.34
1.34 1.32 1.31
K9
1.15 1.18 1.18
1.18 1.16 1.16
1.13 1.10 1.09
K8
1.12 1.11 1.09
1.07 1.04 1.03
0.99 0.97 0.95
K7
1.11 1.08 1.05
1.03 1.00 0.97
0.94 0.92 0.89
K6
1.08 1.05 1.01
0.97 0.95 0.93
0.91 0.88 0.86
K5
1.07 1.02 0.98
0.94 0.92 0.90
0.87 0.85 0.82
K4
1.05 0.99 0.94
0.91 0.88 0.87
0.83 0.82 0.80
K3
1.03 0.96 0.91
0.89 0.86 0.84
0.81 0.80 0.79
QJ
1.17 1.24 1.30
1.34 1.37 1.39
1.38 1.38 1.37
QT
1.15 1.21 1.26
1.28 1.30 1.30
1.30 1.29 1.29
Q9
1.10 1.13 1.13
1.14 1.14 1.12
1.10 1.09 1.07
Q8
1.07 1.06 1.03
1.03 1.01 0.99
0.96 0.94 0.92
Q7
1.03 0.99 0.95
0.93 0.90 0.87
0.85 0.82 0.81
Q6
1.02 0.97 0.93
0.89 0.86 0.83
0.81 0.79 0.76
Q5
1.00 0.94 0.89
0.86 0.83 0.80
0.78 0.76 0.74
Q4
0.98 0.91 0.86
0.83 0.80 0.78
0.75 0.73 0.72
Q3
0.96 0.88 0.84
0.80 0.78 0.75
0.73 0.71 0.70
Q2
0.94 0.86 0.81
0.77 0.74 0.72
0.71 0.69 0.69
JT
1.11 1.17 1.23
1.27 1.28 1.30
1.30 1.31 1.31
J9
1.07 1.09 1.12
1.13 1.12 1.12
1.11 1.10 1.09
J8
1.03 1.03 1.03
1.02 1.00 0.98
0.97 0.95 0.95
J7
0.99 0.95 0.94
0.91 0.89 0.87
0.85 0.83 0.81
J6
0.96 0.89 0.86 0.83
0.79 0.77 0.75
0.72 0.70
J5
0.94 0.87 0.83
0.80 0.76 0.74
0.72 0.69 0.68
J4
0.92 0.85 0.80
0.77 0.74 0.71
0.69 0.67 0.66
J3
0.91 0.82 0.78
0.74 0.71 0.68
0.67 0.65 0.63
J2
0.89 0.79 0.74
0.71 0.69 0.67
0.65 0.63 0.62
T9
1.04 1.06 1.10
1.13 1.13 1.14
1.13 1.13 1.14
T8
0.99 1.01 1.02
1.02 1.01 1.00
1.00 0.99 0.99
T7
0.96 0.94 0.92
0.91 0.90 0.89
0.88 0.87 0.85
T6
0.92 0.87 0.85
0.82 0.80 0.78
0.77 0.75 0.73
T5
0.88 0.81 0.77
0.74 0.72 0.69
0.67 0.65 0.64
T4
0.88 0.80 0.75
0.72 0.69 0.66
0.64 0.63 0.60
T3
0.86 0.77 0.73
0.68 0.66 0.64
0.63 0.61 0.59
T2
0.83 0.74 0.69
0.66 0.64 0.62
0.60 0.59 0.58
98
0.96 0.98 1.00
1.01 1.00 1.00
0.99 0.98 0.98
97
0.92 0.92 0.92
0.91 0.90 0.90
0.89 0.88 0.89
96
0.90 0.86 0.84
0.82 0.81 0.79
0.78 0.77 0.78
95
0.85 0.79 0.76
0.73 0.71 0.70
0.68 0.67 0.66
94
0.81 0.73 0.69
0.66 0.63 0.60
0.59 0.57 0.56
93
0.80 0.72 0.66
0.64 0.60 0.58
0.57 0.55 0.54
92
0.78 0.69 0.65
0.61 0.58 0.56
0.54 0.53 0.52
87
0.90 0.91 0.92
0.92 0.91 0.91
0.91 0.92 0.92
86
0.87 0.86 0.85
0.84 0.83 0.82
0.82 0.82 0.83
85
0.83 0.79 0.77
0.76 0.74 0.73
0.72 0.73 0.72
84
0.79 0.73 0.70
0.67 0.65 0.64
0.62 0.62 0.61
83
0.74 0.67 0.63
0.59 0.57 0.55
0.54 0.52 0.51
82
0.74 0.66 0.61
0.58 0.55 0.53
0.52 0.50 0.49
76
0.84 0.85 0.85
0.85 0.85 0.85
0.86 0.87 0.87
75 0.81 0.80
0.79 0.77 0.77
0.76 0.78 0.78
0.79
74
0.77 0.73 0.71
0.69 0.68 0.68
0.68 0.68 0.68
73
0.73 0.67 0.64
0.61 0.60 0.59
0.58 0.58 0.57
72
0.70 0.62 0.57
0.54 0.52 0.50
0.50 0.48 0.48
65
0.80 0.81 0.80
0.80 0.79 0.80
0.81 0.83 0.84
64
0.76 0.74 0.73
0.72 0.72 0.73
0.73 0.74 0.76
63
0.72 0.69 0.66
0.64 0.63 0.63
0.64 0.64 0.65
62
0.68 0.62 0.59
0.56 0.55 0.54
0.54 0.54 0.54
54
0.76 0.76 0.75
0.75 0.76 0.77
0.80 0.80 0.82
53
0.73 0.70 0.69
0.68 0.69 0.69
0.71 0.72 0.74
52
0.69 0.65 0.62
0.60 0.60 0.61
0.61 0.62 0.62
43
0.70 0.67 0.66
0.65 0.64 0.65
0.67 0.67 0.69
42
0.66 0.62 0.59
0.58 0.57 0.58
0.58 0.59 0.60
32
0.64 0.60 0.56
0.54 0.54 0.54
0.54 0.56 0.56
AK* 1.34 1.53
1.66 1.78 1.86
1.92 2.00 2.04
2.07
AQ* 1.33 1.49
1.60 1.68 1.76
1.80 1.85 1.89
1.92
AJ* 1.32 1.45
1.53 1.61 1.67
1.72 1.76 1.80
1.82
AT* 1.29 1.42
1.48 1.55 1.60
1.65 1.68 1.71
1.74
A9* 1.26 1.33
1.39 1.42 1.45
1.47 1.49 1.52
1.53
A8* 1.24 1.30
1.33 1.36 1.40
1.43 1.44 1.46
1.48
A7* 1.22 1.28
1.30 1.32 1.35
1.37 1.41 1.41
1.43
A6* 1.20 1.23
1.25 1.26 1.29
1.33 1.33 1.37
1.40
A5* 1.19 1.25
1.26 1.30 1.33
1.37 1.40 1.42
1.43
A4* 1.17 1.22
1.24 1.26 1.29
1.34 1.36 1.39
1.42
A3* 1.17 1.18
1.22 1.23 1.27
1.30 1.34 1.38
1.40
A2* 1.15 1.17
1.18 1.21 1.23
1.27 1.30 1.33
1.34
KQ* 1.27 1.40
1.52 1.62 1.70
1.76 1.80 1.83
1.87
KJ*
1.25 1.38 1.47
1.55 1.62 1.67
1.70 1.74 1.77
KT* 1.23 1.34
1.43 1.51 1.55
1.59 1.63 1.65
1.69
K9* 1.21 1.28
1.32 1.38 1.39
1.43 1.44 1.46
1.49
K8* 1.16 1.21
1.23 1.26 1.29
1.30 1.32 1.33
1.36
K7* 1.15 1.17
1.20 1.22 1.24
1.26 1.27 1.30
1.31
K6* 1.13 1.16
1.16 1.18 1.19
1.22 1.23 1.27
1.28
K5* 1.11 1.13
1.13 1.15 1.17
1.20 1.21 1.24
1.26
K4* 1.09 1.09
1.11 1.12 1.14
1.16 1.20 1.22
1.23
K3* 1.08 1.07
1.07 1.09 1.13
1.14 1.15 1.19
1.20
QJ* 1.21 1.31
1.42 1.49 1.57
1.64 1.65 1.70
1.72
QT* 1.19 1.29
1.37 1.45 1.51
1.55 1.59 1.64
1.66
Q9* 1.15 1.22
1.27 1.31 1.36
1.39 1.41 1.43
1.46
Q8* 1.12 1.14
1.18 1.23 1.25
1.27 1.30 1.30
1.33
Q7* 1.09 1.10
1.11 1.12 1.14
1.16 1.19 1.20
1.19
Q6* 1.07 1.07
1.07 1.10 1.10
1.13 1.14 1.16
1.18
Q5* 1.06 1.05
1.06 1.06 1.08
1.10 1.11 1.13
1.15
Q4* 1.04 1.02
1.01 1.04 1.06
1.07 1.09 1.12
1.13
Q3* 1.02 1.01
1.00 1.02 1.03
1.06 1.08 1.10
1.12
Q2* 1.00 0.98
0.97 0.98 1.01
1.03 1.06 1.08
1.10
JT* 1.15 1.26
1.35 1.43 1.49
1.54 1.59 1.62
1.66
J9* 1.11 1.18
1.25 1.30 1.34
1.37 1.40 1.43
1.46
J8* 1.07 1.13
1.16 1.20 1.23
1.25 1.28 1.30
1.33
J7* 1.04 1.07
1.07 1.10 1.13
1.15 1.16 1.19
1.20
J6* 1.02 1.01
1.01 1.01 1.04
1.05 1.07 1.09
1.11
J5* 0.99 0.98
0.98 1.00 1.02
1.02 1.03 1.06
1.07
J4* 0.98 0.95
0.96 0.97 0.99
1.00 1.01 1.04
1.06
J3* 0.95 0.93
0.93 0.94 0.97
0.99 1.01 1.03
1.04
J2* 0.96 0.91
0.91 0.92 0.94
0.96 0.98 1.01
1.03
T9* 1.08 1.16
1.24 1.30 1.35
1.38 1.42 1.46
1.49
T8* 1.04 1.10
1.16 1.19 1.24
1.27 1.29 1.32
1.35
T7* 1.01 1.05
1.09 1.10 1.13
1.16 1.19 1.20
1.24
T6* 0.98 0.99
1.00 1.02 1.04
1.07 1.08 1.11
1.13
T5* 0.94 0.93
0.93 0.94 0.96
0.98 0.99 1.01
1.04
T4* 0.94 0.90
0.91 0.92 0.94
0.95 0.97 0.98
1.00
T3* 0.91 0.88
0.88 0.90 0.91
0.93 0.96 0.97
0.98
T2* 0.90 0.86
0.86 0.87 0.89
0.92 0.93 0.95
0.98
98* 1.01 1.08
1.13 1.18 1.22
1.24 1.27 1.30
1.34
97* 0.98 1.01
1.06 1.10 1.13
1.16 1.19 1.22
1.24
96* 0.94
0.95 1.00 1.03
1.04 1.07 1.10
1.13 1.14
95* 0.91 0.90
0.92 0.94 0.95
0.97 0.99 1.01
1.04
94* 0.88 0.86
0.86 0.87 0.87
0.89 0.91 0.93 0.94
93* 0.87 0.84
0.84 0.84 0.86
0.87 0.89 0.91
0.93
92* 0.85 0.81
0.81 0.82 0.84
0.85 0.87 0.89
0.91
87* 0.96 1.01
1.06 1.10 1.14
1.18 1.21 1.23
1.27
86* 0.93 0.97
1.00 1.03 1.06
1.09 1.12 1.15
1.19
85* 0.90 0.89
0.92 0.95 0.97
1.01 1.04 1.05
1.08
84* 0.85 0.85
0.85 0.87 0.89
0.92 0.94 0.96
0.97
83* 0.82 0.79
0.80 0.81 0.82
0.83 0.84 0.87
0.90
82* 0.81 0.77
0.78 0.77 0.80
0.82 0.84 0.86
0.88
76* 0.91 0.95
1.00 1.02 1.06
1.12 1.15 1.18
1.22
75* 0.88 0.90
0.95 0.97 1.00
1.03 1.09 1.11
1.15
74* 0.84 0.86
0.87 0.90 0.92
0.95 0.99 1.02
1.05
73* 0.79 0.79
0.81 0.82 0.84
0.86 0.90 0.91
0.94
72* 0.75 0.74
0.75 0.76 0.77
0.78 0.81 0.84
0.86
65* 0.87 0.91
0.95 0.98 1.02
1.07 1.10 1.15
1.19
64* 0.83 0.85
0.88 0.91 0.95
1.00 1.03 1.07
1.10
63* 0.80 0.81
0.82 0.84 0.88
0.91 0.94 0.97
1.01
62* 0.75 0.74
0.76 0.77 0.79
0.83 0.85 0.88
0.89
54* 0.83 0.86
0.91 0.94 0.99
1.03 1.07 1.12
1.17
53* 0.79 0.81
0.84 0.88 0.92
0.96 0.99 1.05
1.08
52* 0.76 0.76
0.77 0.81 0.84
0.87 0.92 0.95
0.98
43* 0.78 0.79
0.82 0.85 0.88
0.93 0.96 1.00
1.04
42* 0.73 0.74
0.76 0.78 0.81
0.86 0.89 0.92
0.96
32* 0.71 0.72
0.73 0.75 0.78
0.81 0.85 0.89
0.91