Probability

IMPORTANT NOTE; This article has been written by a guest writer and not by The Mob. It is intended as a rough and ready guide to working out calculations quickly whilst playing poker at the table to enable you to make decisions easier. There are certain calculatons that are not 100% accurate.

Simon writes below;

If you were striking a wager with a friend that at least one of the dice was a 6, then this would bring in the additive laws of probability:

1/6 + 1/6 =1/3 (=33.34% or 2/1)

This is 'close enough' for a quick decision at the table but is in fact incorrect. It is in fact a 30.56% chance. It is often easier to calculate the chances of something not happening and then subtracting that from 1.

ie 5/6 x 5/6 = 25/36 ANS is 36/36 - 25/36 = 11/36 or 30.56%

Nevertheless we believe that Simon has achieved what he set out to do and are very happy to publish his article in full below.

Calculating Poker Probability
By Simon "Fester" Galloway

Introduction

Many of you will have heard an expression such as: "he was a 4.5/1 dog and caught me on the river" most times you have played from the bad beat brigade, but may not fully appreciate how the odds were calculated. The purpose of this article is therefore to give you the tools to be able to calculate such eventualities for yourselves. The benefits of being able to do this are:
1) You will be able to counter-irritate the story teller by correcting his version of the odds (which were probably made up anyway) and:
2) You will be able to calculate for yourself which hands merit investing more of your stack in order to see another card, based on the probability of you winning the hand.

The second reason is obviously of more use to you as poker players, but the former can be handy as well, if for no better reason than to make the bad beat brigade find a less knowledgeable audience.
The purpose of this article is therefore NOT to gain you an 'A' grade at A Level; the answer does not need to be correct to 3 decimal places in order for you to be able to base your decision on sound mathematical probability. Some eventualities are actually quite difficult to work out; you certainly won't have time to get very far with the maths during a hand. It is arguably more in your favour to make quick decisions based on useful approximations than it would be to take an excessive amount of time to think things through as this gives others the opportunity to work out that you are likely behind at that moment in time.
Various methods are available to help you arrive at your answer; I will use general gambling concepts to illustrate these before turning my attention solely to poker. If anyone spots any mistakes, please let me know - it may explain away some of my losses recently!
Also, probability is conventionally expressed as a decimal between 0 and 1. I have converted this figure into imperial odds, which are more widely understood.

Independent Probability

Playing craps, the outcome of each die is independent of the other die and the outcome of 2 dice rolled together can be derived from multiplication laws of probability:

Many scenarios can be evaluated by dividing the number of DESIRABLE outcomes by the number of POSSIBLE outcomes.

So, to find out the true odds of rolling double 6 in a roll:
Only 1 desirable outcome for each die (i.e. a "6") and the events are independent so:

1/6 * 1/6 = 1/36 (=2.78% or 35/1)

You can now judge for yourselves if the 33/1 and 30/1 that UK and US casinos offer against this roll make this a bet worth doing regularly…

If you were striking a wager with a friend that at least one of the dice was a 6, then this would bring in the additive laws of probability:

1/6 + 1/6 =1/3 (=33.34% or 2/1)

Stay with me on this, it is fundamental to understand this difference. The probability of A AND B happening as independent events = A * B. The probability of A OR B happening as independent events = A + B. Re-read that summary, it will be used later!

Note that if Michael Owen is 3/1 to be first scorer and Liverpool to win 1-0 is 6/1, no bookie will offer you the 27/1 double because these events are not independent. This brings in to play another branch of mathematics beyond the scope of this article. However, all you need to realise is that as a hand unfolds more and more cards become known and the remaining deck (i.e. the remaining possibilities) become smaller. It is usually sufficient to ignore this effect to make it far quicker to arrive at a useable approximation. If you have "5 outs twice" you can approximate by adding

5/50 + 5/50 to quickly arrive at a 4/1 shot. (The true answer will be slightly shorter odds because there aren't 50 cards left - but it's good enough to work with!!

So, playing casino stud poker, what is the probability of being dealt any royal flush?
The first card can be any of 20 (T,J,Q,K,A of each suit). Thereafter, there are only 4 possible cards available to continue the hand, then 3 possible cards etc until the last card HAS to be 1 card in particular. Mathematically, this looks like:
{20/52 * 4/51 * 3/50 * 2/49 * 1/48} = 649,739/1 so the house odds of 50/1 look a touch on the slender side! In a private poker game, don't worry about working this one out, just concentrate on extracting the maximum return from the hand!

Combinations and Permutations

How many 2 card starting hands are there in hold em? Lou Krieger suggests ignoring suits at this stage and therefore there are 169 possibilities. This is derived from 13 possibilities of rank on the first card and 13 possibilities on the second. If we include suit possibilities available, we need to calculate how many 2 card possibilities can be chosen from a deck of 52 cards. As AhKh plays exactly the same as KhAh, we are not concerned with order and so we are calculating Combinations. If the order were important, then Permutations would apply.

The horrible Combination formula bit: n items can be chosen r ways at a time:

n! / {(n-r)! * r!}

For 2 card starting hands, this is calculated as:

52! / (50! * 2!)

(! is read as "factorial" where 6! = 6*5*4*3*2*1 and 3! = 3*2*1)

From the above, although it looks frightening to start with, you will quickly realise that through cancelling you can quickly arrive at (52 * 51) / 2 = 1326 possible hands, each with an equal chance of occurring. Yes, that's right! You have as much chance of being dealt AcAs as you have of being dealt 7d2h - it just doesn't feel like it!

So, how many trebles are possible if you choose 6 horses? Again the order is unimportant, so there are 6! / (3! * 3!) possible ways. This breaks down to be 720 / 36, i.e. 20 trebles.
You can also use Pascal's triangle to get this answer (Pascal the French mathematician, not Pascal the poker player!!) If you back horses in multiple bets a lot and would like to know more, mail me!

Texas Holdem

OK, we now have a few tools to help us calculate various eventualities. Exactly which tool to use will vary from situation to situation. Unfortunately only practice will allow you to choose the best way first time every time.

Lets take a tournament scenario: Shortstacked, you go all in with JT looking to pick up the blinds, but get called by the big blind who has A8. The cards are turned "on their backs" to see the flop come A 8 3. Even the non-mathematicians will realise that it is now time to stand up and order a taxi - if you haven't already done so! But, what is the probability that you get lucky? Well, in words you need to work out the probability of catching either:

1) TT (to give you trips)
2) JJ (to give you trips)
3) Q9, 9Q, 97 or 79 (to give you a straight)
Any other outcome is the end of your evening. An A or an 8 would add unnecessary insult to injury by completing a full house for your opponent.
As you can see, by the time you work out the answer, you will be half way home!

1) The likelihood of TT can be calculated from (3/45 * 2/44) = 329/1
[Note that we know what 7 of the cards are, so only 45 possibilities remain on the turn, and 44 on the river]

2) The likelihood of JJ will be the same as TT = 329/1

3) The likelihood of the straight will be (16/45 * 4/44) = approx. 30/1

So, the probability of getting lucky will be the probability of 1) OR 2) OR 3) happening.
This equals (1/330 + 1/330 + 1/31) = approx. 25/1 = Taxi!!

So, why have I made you wade through that? Well, if you coped with that, the following scenarios are all regular questions asked in poker circles that you now have the tools to answer:

Q If I flop a 4-flush, what are my odds of making the flush?
A (9/47 + 9/46) = 1.58/1 (I've seen a figure of 1.86/1 used elsewhere. It matters not, approximate by calculating 9/45 + 9/45 (1/5 + 1/5) and you will quickly see that you are roughly a 40% chance or 1.5/1 shot.

Q If I missed on the turn, what are the odds of making that flush now?
A (9/46) = 4.11/1 (At the table, you can either remember that or identify that the fraction is very similar to 9/45 which would be a 4/1 shot. So, you can readily work out that you are a little worse than a 4/1 shot. The 0.11 difference is unlikely to affect your decision, even if you can be bothered to work it out!)

Q If I hold a small pair, how often can I expect to hit trips?
A (2/50 + 2/49 + 2/48) = 7.16/1 (Again, I have seen 7.5/1 commonly quoted, but it is close enough. In fact, if you rounded the figures to (2/50 + 2/50 + 2/50) you could deduce that 6 chances in 50 give you somewhere between a 7/1 and 7.5/1 shot and that is as close as you need to know.

Q (taken from LNP4) Simon Trumper asked Phil Hellmuth what odds he was getting after deciding (rather wisely) to pass Q2 against QQ. After a brief thought, Phil Hellmuth replied "no worse than 8/1" My instant reaction was that I thought it would be higher than that, but in fact the answer was stunningly accurate. There are lots of outcomes that need to be taken into consideration and so an exact answer would take some thinking about. Possible outcomes include:

· Making a straight with the 2
· Making a straight with the Q (split pot)
· Making a flush - either player
· The case Q hitting the board
· Catching 2 or more 2's

A Maybe when I have a couple of days spare I will tackle it! I think this is one of those situations best committed to memory - if you feel the need to work it out, let me know how you get on! I've had 2 attempts and made it between 7.5/1 and 8/1 both times, but slightly different on each occasion…

Summary and Conclusion

OK, so quite a lot of information to deal with in a short space of time, what can we take away and use at the tables?

1. Rather than memorise a table of odds, you can now quickly work out your odds to the nearest point of common situations at the poker table.
2. Remember to work with pot odds and implied pot odds in conjunction with the probability of your hand improving when deciding when to continue. Plenty written on this subject - just remember that most calculations don't allow for you to hit your hand AND STILL LOSE! So, if you are chasing a flush and the board is paired, for example, you need to make an adjustment so that the reward of winning the pot now compensates for the occasion when your flush runs into a full house.
3. Understanding probability is one of the many variables that form a good poker player. It can save you money over the year by stopping you chasing unrewarding dreams. However, don't rely on it too much - all the other variables are collectively far more valuable than being numerate.
4. I have tried to illustrate how to switch between statistical theory and approximate fractions that are easy to work with. Although mathematicians may be groaning at the sacrilege, it is necessary in order to give the average poker player anything like enough time to work things out.
5. Anyone who would like to challenge this article or improve it is most welcome. If on the off chance you find this helps your game, beers are always gratefully accepted! Now what are the odds on that happening???

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