Wild Viking

We deal with a finite number of possibilities as is with every card game, because there are only a set number of cards. As there are a finite number of cards we can relatively easily calculate the chances or odds of getting a particular hand. All we have to look at are the combinations that can be obtained three-card, two-card or even one-card ones on the board. Also we need to take into account the ones you can make but also your opponents can make, these are called favorable combinations. The formula we are going to apply needs to count the favorable combinations that are prone to happen. We need a lot of information to do this calculation, the number of cards showing on the board, the number of opponents and the cards they hold, as well as the cards you have. We ignore information like seeing opponents’ cards, removing or adding cards from/to the deck or specific shuffling so cards are always dealt in order. Texas Hold ‘Em poker has a lot of chance calculus, because the chances change so much per game. The odds of getting a card fluctuate enormously between to extremes.

There are the odds that are named the long shot odds. This is due to that everything happens in set order, and the information you get as well. It consists of own hand probabilities and opponents’ hand probabilities. The first is only calculated after the flop and the turn, the latter is calculated post-flop, post-turn and post-river (All of this means after as well).
There are also the odds that are called immediate odds, these are the odds that can be calculated always without needing events to happen in set order. Examples are pre-flop, turn odds, and the odds to get that combination you want.
The long shot odds are there to give you insight when confronted to make a decision and the immediate odds are there to give you an advantage in the betting rounds. Especially the long shot odds are important, as that will make you walk away with the money every time or not. Now it is not like the immediate odds are to be forgotten, it can make your opponent fold even in the case that he has a better hand. This is the psychological element in poker, the element that makes it exciting.

Let’s take a more in depth look at immediate odds:
- preflop odds ; the calculation of getting a good initial hand
- flop odds ; after getting the two cards but before the three cards come on the board.
- turn odds ; after getting the flop, the chance of getting a good fourth card.
- river odds ; after getting the turn, the chance of getting a good river card.
There are just too many odds to calculate so let’s consider a few cases.
We will start with preflop odds.
The chance you will get pocket rockets or a pair of Aces, which is the same.
Favorable combinations : C ( 4 above 2). Possible combinations : C ( 52 above 2) = 1326.
P = Favorable combinations / Possible combinations.
P = C ( 4 above 2) / C ( 52 above 2) = 0,452% (roughly 220 to one)
This basically hold for any pair.
So to get two Aces or two Kings (two cowboys) you add the two together, because the chances of both of them are the same.
P = 2*( C ( 4 above 2) / C ( 52 above 2) ) = 0,904% (roughly 110 to one)
This goes for every two pair you are comparing so we can calculate the possibility of getting any pair.
P = 13 * ( C ( 4 above 2) / C ( 52 above 2) ) = 5.882% (roughly 17 to one). Why thirteen times that number, well we have thirteen pairs in total: from the two to the Ace, that are thirteen cards.
The chance you will get two suited cards.
For any suit (spades, clubs, diamonds and hearts) we get C (13 above 2) = 78 combinations that are favorable. The actual number is 78*4 (suites) = 312. P = 4 * ( C ( 13 above 2) / C ( 52 above 2) ) = 23,529 % (roughly 4 to one).
Now for some more advanced stuff, we have a pair and what are the chances someone else has a higher pair. We start of by saying we have any pair lower that a pair of Aces. We need to define our variables which are; n meaning number of your opponents, p meaning the number of pairs with a higher value than your pair. Example: You hold a pair of eights, then there are only six pairs above you namely nines,tens,jacks,queens,kings and aces so p = 6. If you have two kings then p=1 for only Aces are above you in value.
The formula is as follow:
P = ((6n*p)/1225) – (C(n above 2)*p(6p – 1)/230300) + (C(n above 3)*p(6p – 1)*((6p – 2)/238360500)) = ((6n*p)/1225) – (n*(n – 1)*(p(6p – 1)/460600)) + n*((n – 1)(n – 2))*p(6p – 1)(6p – 2)/1430163000
Here are the numbers for when you are holding kings (p = 1) and n ranges from 1 to 9.
P = 0,489% for n=1, P = 0,979% for n=2, P = 1,462% for n=3, P = 1,946% for n=4, P = 2,427% for n=5, P = 2,906% for n=6, P = 3,383% for n=7, P = 3,912% for n=8, P = 4,330% for n=9 .
So on an normal poker table with a total of six players, the odds that when you have a pair of kings any opponent will have a pair of Aces is roughly one in 25.
Now some chances that a specific opponent (n =1) will have a higher pair for every pair you could have.
Pair of twos P = 5,877% , Pair of threes P = 5,387%, Pair of fours P = 4,897%, Pair of fives P = 4,408%, Pair of sixes P = 3,918%, Pair of sevens P = 3,428%, Pair of eights P = 2,938%, Pair of nines P = 2,448%, Pair of tens P = 1,959%, Pair of jacks P = 1,469%, Pair of queens P = 0,979%, Pair of kings P = 0,489%.
We move on to calculate some flop odds. All we know is the two cards we can see, which leaves 50 unknown cards that can create ( C ( 50 above 3) ) 19600 possible combinations for the board that will show up on the flop.
The chances of getting a three-of-a-kind or four-of-a-kind
We name C the card that we are expecting and we name c the number or cards you hold with value C. c can only be none,one or two. The cards with value C are calculated with 4-c. So here are the numbers for 0 c;
one C: P = 21.122%. two C: P = 1.408%. three C: P = 0.020%.
for 1 c;
one C: P = 16.545%. two C: P = 0.719%. three C: P = 0.005%.
for 2 c;
one C: P = 11.510%. two C: P = 0.244%. three C: P = 0% (there aren’t five cards in one deck).
So if you hold a pair the chance to get a three-of-a-kind is 11,510%.
The chance of getting a flush
We name S to be the suit and s is the number of cards of suit S. s can only be none, one or two, and the cards that are left of suit S is 13 – s.
Next are the values or chances for 0 s;
one S: P = 44,173%, two S: P = 14,724%, three S: P = 1,459%.
for 1 s;
one S: P = 43,040%, two S: P = 12,795%, three S: P = 1,122%.
for 2 s;
one S: P = 41,586%, two S: P = 10,943%, three S: P = 0,841%.
So we have two suited cards ( P = 23,529 %) and we want a flush on the flop the chance of that would be 0,841%.