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.


	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 44 = 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 129 = 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 = 4C(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: 46 = 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 411 = 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 1211 = 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 412 = 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 1212 = 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: 6n/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/(12251128). 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’ = 6n/1225 – (n(n-1)/2)(6/(12251128)) =… = *((2456n – 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.