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Texas Hold'em Preflop Odds With Supporting Math

1) AA 
The event is “AA dealt on your hand”.
Favorable combinations: C(4,2) = 6 (two aces from all four)
Possible combinations: C(52,2) = 1326
The probability is P = C(4,2)/C(52,2) = 6/1326 = 0.452% (221 to 1)

2) AA or KK
These two events are incompatible, so their odds will be added.
Each of them has the probability C(4,2)/C(52,2) (see 1) AA), so
P = 2* C(4,2)/C(52,2) = 12/1326 = 0.904% (110.5 to 1)

3) KK, QQ, JJ
Similarly, we have P = 3* C(4,2)/C(52,2) = 18/1326 = 1.357% (73.66 to 1)

4) 10 10, 99, 88, 77, 66
Similarly, we have P = 5* C(4,2)/C(52,2) = 30/1326 = 2.262% (44.2 to 1)

5) 55, 44, 33, 22
Similarly, we have P = 4* C(4,2)/C(52,2) = 24/1326 = 1.809% (55.25 to 1)

6) Pair (Any)
We have 13 pair types and for each type the probability is C(4,2)/C(52,2).
Then P = 13* C(4,2)/C(52,2) = 78/1326 = 5.882% (17 to 1)

7) AK Suited
For a specific symbol, we have one single (AK) suited combination. We have four symbols, so 4 such favorable combinations.
P = 4/C(52,2) = 4/1326 = 0.301% (331.5 to 1)

8) AK Offsuit
We have 4*4 = 16 of (AK) combinations. By subtracting the 4 suited (see 7) AK suited), we find 12 favorable combinations.
P = 12/C(52,2) = 12/1326 = 0.904% (110.5 to 1)

9) AQ or AJ Suited
These two events are incompatible, so we can add their odds.
For AQ suited, we have 4 favorable combinations and also 4 for AJ (see 7) AK suited), so totally 8 favorable combinations.
P = 8/C(52,2) = 8/1326 = 0.603% (165.75 to 1)

10) AQ or AJ Offsuit
For AQ offsuit we have 12 favorable combinations and also 12 for AJ offsuit (see 8) AK offsuit), totally 24 favorable combinations.
P = 24/C(52,2) = 24/1326 = 1.809% (55.25 to 1)

11) KQ Suited
As it in 7) AK suited case, P = 4/C(52,2) = 4/1326 = 0.301% (331.5 to 1)

12) KQ Offsuit
As it in 8) AK offsuit case, P = 12/C(52,2) = 12/1326 = 0.904% (110.5 to 1)

13) A and <J Suited
Favorable combinations: (A2), (A3), (A4), (A5), (A6), (A7), (A8), (A9), (A 10) suited.
Each of the 9 types has 4 combinations, so totally we have 36.
P = 36/C(52,2) = 36/1326 = 2.714% (36.83 to 1)

14) A and <J Offsuit
Each of above favorable combination type has 12 combinations (see 8) AK offsuit), so totally we have 12*9 = 108 favorable combinations.
P = 108/C(52,2) = 108/1326 = 8.144% (12.27 to 1)

15) Two Cards Suited
For a specific symbol, we have C(13,2) = 78 favorable combinations. Multiplying by 4 (symbols), we find 312 favorable combinations.
P = 4*C(13,2)/C(52,2) = 312/1326 = 23.529% (4.25 to 1)

16) A or a Pair
The two events whose probability we are looking for are:
“(Ax) dealt, x any card” and “(yy) dealt, y any card”
These are not incompatible events (there exist common favorable combinations – namely (AA)).
Therefore, the probability to be dealt A or pair is P = P(“(Ax) dealt, x any card”) + P(“(yy) dealt, y any card”) – P(“(AA) dealt). (*)
For (Ax):
(Ax), x different from A: 4*(52-4) = 192 combinations
(AA): C(4,2) = 6 combinations
Totally, we have 198 favorable combinations for (Ax).
For (yy): 78 favorable combinations (see 6) pair (any))
By replacing all these in the (*) formula, we find:
P = 198/1326 + 78/1326 – 6/1326 = 270/1326 = 20.361% (4.91 to 1)

17) Any Two >J
Favorable combinations: (xy), with x, y from the set (Q, K, A).
(QK): 16
(KA): 16
(QA): 16
Totally, we have 48 favorable combinations.
P = 48/C(52,2) = 48/1326 = 3.619% (27.625 to 1)

18) 72 Offsuit 
(72): 16
(72) suited: 4
Favorable combinations: 16 – 4 = 12
P = 12/C(52,2) = 12/1326 = 0.904% (110.5 to 1)

19) AA, KK, QQ, JJ
Favorable combinations: 4*6 = 24 (see 1) AA)
P = 24/C(52,2) = 24/1326 = 1.809% (55.25 to 1)

20) A In Your Hand
Favorable combinations: (Ax) , in number of 198 (see 16) A or pair)
P = 198/C(52,2) = 198/1326 = 14.932% (6.69 to 1)

21) A High
Favorable combinations: (Ax), with x different from A, in number of 192 (see 16) A or pair)
P = 192/C(52,2) = 192/1326 = 14.479% (6.90 to 1)

22) Suited Connectors (23, KQ)
We have 11 such suited connectors: (23), (34), (45), (56), (67), (78), (89), (9 10), (10, J), (JQ), (QK). Each type has 4 combinations (see 7) AK suited), so totally we have 4*11 = 44 favorable combinations.
P = 44/C(52,2) = 44/1326 = 3.318% (30.13 to 1) 

23) Connectors Offsuit 
Each of above offsuit combination type has 12 combinations (see 8) AK offsuit), so we totally have 12*11 = 132 favorable combinations.
P = 132/C(52,2) = 132/1326 = 9.954% (10.04 to 1)

24) Two Adjacent & Suited
We have 12 such combination types: (23), (34), (45), (56), (67), (78), (89), (9 10), (10, J), (JQ), (QK), (KA) and each of them has 4 combinations, so totally we have 4*12 = 48 favorable combinations.
P = 48/C(52,2) = 48/1326 = 3.619% (27.625 to 1)

25) Two Adjacent & Offsuit
Each of above types has 12 offsuit combinations, so totally we have 12*12 = 144 favorable combinations.
P = 144/C(52,2) = 144/1326 = 10.859% (9.20 to 1)

26) AA Up Against a KK
The probability for one specific opponent to be dealt KK, if you were dealt AA:
Favorable combinations (for that event to occur): C(4,2) = 6
Possible combinations: C(50,2) = 1225.
The probability is then 6/1225.
Let n be the number of your opponents.
Maximum 2 opponents could simultaneously be dealt (KK).
The probability for at least one opponent to be dealt KK will be:
6*n/1225 – C(n,2)*P(“two opponents simultaneously are dealt KK”)
Let’s find this last probability:
Favorable double combinations: ((KK)(KK)), in number of C(4,2)*C(2,2) = 6.
Probability will be 6/(C(50,2)*(C(48,2)) = 6/(1225*1128).
So, the final results are:

a) The probability for one opponent (a fixed one) to be dealt KK while you are dealt AA is
6/1225 = 0.489% (204.1 to 1)

b) The probability for at least one opponent to be dealt KK while you are dealt AA is
P’ = 6*n/1225 – (n*(n-1)/2)*(6/(1225*1128)) =… = ((2456*n – n*(n-1))/460600), where n is the number of your opponents.
(For a full table, n=9, the returned result is 4.783%).
You may generate a table of values for different values of n.
Untitled Document
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QUICK REFERENCE ODDS
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POT ODDS -PART 2
POKER TOURNAMENT TIPS
BLUFFING IN POKER
ONLINE POKER TELLS?
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PLAYING SHORT HANDED
PLAYER PROFILES
THE WORST HAND IN POKER
POKER BY NUMBERS
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POCKET PAIRS
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COMMON HOLDEM MISTAKES
CREATING TABLE IMAGE
BASIC BETTING STRATEGY
OMAHA POKER STRATEGY
CHANGING PACE IN POKER
BIG SUITED CONNECTORS
UNSUITED CONNECTORS
LIMITS IN NO-LIMIT HOLDEM
PLAYING PAST THE FLOP
FREE CARDS
VALUE BETTING IN POKER
TABLE SELECTION IN POKER
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