Know Your Chances

Texas Hold'em Poker Probability/Statistics for Single Hand Pocket Cards

In this section we will be demonstating the possible outcomes for pocket card hands in Texas Hold'Em. We will run through the various possible scenarios the player may have.

Scenario 1

In this scenario we will be examining the best hand in Hold'Em.

A SA D   BlankBlankBlankBlankBlank

Hand NameAceKingQueenJack10OtherTotalAccum.
Straight Flush0.0044%0.0001%0.0002%0.0002%0.0002%0.0051%0.0102%0.0102%
Four of a Kind0.8163%0.0022%0.0022%0.0022%0.0022%0.0174%0.8424%0.8526%
Full House6.4045%0.1786%0.1786%0.1786%0.1786%1.4288%8.5476%9.4002%
Three of a Kind11.7738%0%0%0%0%0%11.7738%24.3541%
Two Pairs39.6674%0%0%0%0%0%39.6674%64.0214%
Single Pair35.9786%0%0%0%0%0%35.9786%100%
High Card0%0%0%0%0%0%0%100%

Total Possibilities


The main points of interest in this scenario are:
1. Almost a 1 in 3 chance of having just a pair of aces at the end.
2. Almost a 1 in 4 chance of having three of a kind or better.
3. Almost a 1 in 10 chance of having a full house or better.
4. Less than 1% chance of having a four of a kind.

Scenario 2

In this scenario we will be examining a pocket pair of Jacks.

J HJ S   BlankBlankBlankBlankBlank

Hand NameAceKingQueenJack10OtherTotalAccum.
Straight Flush0.0044%0.0043%0.0043%0.0043%0.0001%0.0009%0.0185%0.0185%
Four of a Kind0.0022%0.0022%0.0022%0.8163%0.0022%0.0174%0.8424%0.8609%
Full House0.1952%0.1952%0.1952%6.3547%0.1786%1.4288%8.5476%9.4085%
Three of a Kind0%0%0%11.726%0%0%11.726%25.0575%
Two Pairs3.6735%3.5877%3.5019%28.7616%0%0%39.5246%64.5821%
Single Pair0%0%0%35.4179%0%0%35.4179%100%
High Card0%0%0%0%0%0%0%100%

Total Possibilities


The main point of interest in this scenario is:
1. Probabilties very similar to 2 Aces except tilted towards the Jack.

Scenario 3

In this scenario we will be examining a pocket Ace and King off-suit.

A SKH   BlankBlankBlankBlankBlank

Hand NameAceKingQueenJack10OtherTotalAccum.
Straight Flush0.0044%0.0022%0.0001%0.0002%0.0002%0.003%0.0102%0.0102%
Four of a Kind0.051%0.051%0.0022%0.0022%0.0022%0.0174%0.1259%0.1361%
Full House0.8006%0.7819%0.0583%0.0583%0.0583%0.4667%2.2241%2.3602%
Three of a Kind1.4707%1.4707%0.1329%0.1329%0.1329%1.0874%4.4275%12.051%
Two Pairs12.4163%6.8984%0.5981%0.5437%0.4893%1.7127%22.6585%34.7095%
Single Pair11.3325%11.3325%1.9981%1.9981%1.9981%16.9129%45.5723%80.2819%
High Card19.7181%0%0%0%0%0%19.7181%100%

Total Possibilities


The main points of interest in this scenario are:
1. Just a 1 in 5 chance of ending on a high card (which has to be an Ace).
2. A 45.6% chance of ending with a single pair.
3. Just over a 1 in 3 chance of ending with a hand better than a single pair.

Scenario 4

This is the same scenario as above except with same-suit cards.

A DK D   BlankBlankBlankBlankBlank

Hand NameAceKingQueenJack10OtherTotalAccum.
Straight Flush0.0512%0.0001%0.0001%0.0002%0.0002%0.003%0.0548%0.0548%
Four of a Kind0.051%0.051%0.0022%0.0022%0.0022%0.0174%0.1259%0.1808%
Full House0.8006%0.7819%0.0583%0.0583%0.0583%0.4667%2.2241%2.4049%
Three of a Kind1.4629%1.4629%0.1267%0.1267%0.1267%1.0365%4.3424%16.3663%
Two Pairs12.246%6.759%0.5607%0.5097%0.4588%1.6057%22.1399%38.5062%
Single Pair10.9992%10.9992%1.8554%1.8554%1.8554%15.7048%43.2695%81.7757%
High Card18.2243%0%0%0%0%0%18.2243%100%

Total Possibilities


The main point of interest in this scenario is:
1. A better chance of obtaining a flush compared to the off-suit cards, there is now a 6.5% chance.

Scenario 5

In this scenario we will examine a matching pair lower down in the deck.

9 C10 C   BlankBlankBlankBlankBlank

Hand NameAceKingQueenJack10OtherTotalAccum.
Straight Flush0.0023%0.049%0.049%0.049%0.049%0.0008%0.1991%0.1991%
Four of a Kind0.0022%0.0022%0.0022%0.0022%0.051%0.0662%0.1259%0.325%
Full House0.0595%0.0595%0.0595%0.0595%0.7983%1.188%2.2241%2.5492%
Three of a Kind0.1296%0.1267%0.1238%0.1209%1.4362%2.3258%4.2629%21.7684%
Two Pairs1.9625%1.8558%1.7556%1.6617%9.1753%5.3547%21.7657%43.5341%
Single Pair1.9217%1.8223%1.7229%1.6235%10.3139%23.1693%40.5735%84.1077%
High Card7.9462%4.1026%2.2025%1.0796%0.5614%0%15.8923%100%

Total Possibilities


The main points of interest in this scenario are:
1. A slighly higher chance of getting a decent hand as the scenario above except with lower value cards.
2. A greater chance of obtaining a straight.

From this, the previous scenario and the mathimatical probabilites in cards we can ascertain that there is approximately a 1.5% chance of obtaining a flush in any pocket hand which is off suit and there is approximately a 6.5% to 7% of obtaining a flush when the pocket cards are of a matching suit.

Scenario 6

In this scenario we will examine pocket cards which are spaced out but still able to make a straight.

10K C   BlankBlankBlankBlankBlank

Hand NameAceKingQueenJack10OtherTotalAccum.
Straight Flush0.0044%0.0043%0.0022%0.0023%0.0023%0.0009%0.0164%0.0164%
Four of a Kind0.0022%0.051%0.0022%0.0022%0.051%0.0174%0.1259%0.1423%
Full House0.0595%0.8%0.0589%0.0589%0.7802%0.4667%2.2241%2.3665%
Three of a Kind0.1329%1.4617%0.1299%0.1299%1.4617%1.0844%4.4005%14.4828%
Two Pairs1.9857%11.6044%1.1544%1.1017%4.9717%1.7127%22.5306%37.0134%
Single Pair1.9981%11.05%1.8733%1.8554%11.05%16.6096%44.4366%81.45%
High Card8.3639%10.1861%0%0%0%0%18.55%100%

Total Possibilities


The main points of interest in this scenario are:
1. There is a drop in the chance of obtaining a straight.
2. As expected, there is a drop in the probability of a flush.

Scenario 7

In this scenario we will examine pocket cards which are spaced out far enough that it isn't possible for both cards to be used in a straight.

7 SQ H   BlankBlankBlankBlankBlank

Hand NameAceKingQueenJack10OtherTotalAccum.
Straight Flush0.0023%0.0023%0.0023%0.0022%0.0023%0.0071%0.0185%0.0185%
Four of a Kind0.0022%0.0022%0.051%0.0022%0.0022%0.0662%0.1259%0.1444%
Full House0.0595%0.0595%0.7994%0.0589%0.0589%1.188%2.2241%2.3685%
Three of a Kind0.1359%0.1359%1.4796%0.1359%0.1359%2.4311%4.4545%11.8261%
Two Pairs2.0262%1.9548%10.9721%1.1354%1.0692%5.6287%22.7864%34.6125%
Single Pair2.123%2.1052%11.4031%2.0874%2.0695%26.0681%45.8563%80.4688%
High Card9.2049%5.4202%4.9062%0%0%0%19.5312%100%

Total Possibilities


The main point of interest in this scenario is:
1. The probability of a straight takes a hit compared to the previous scenario while the other probabilities remain roughly the same.

Scenario 8

For interest sake, in this scenario we will examine the worst pocket cards in Hold'Em.

2 S7 D   BlankBlankBlankBlankBlank

Hand NameAceKingQueenJack10OtherTotalAccum.
Four of a Kind0.0022%0.0022%0.0022%0.0022%0.0022%0.1151%0.1259%0.1423%
Full House0.0595%0.0595%0.0595%0.0595%0.0595%1.9268%2.2241%2.3664%
Three of a Kind0.1359%0.1359%0.1359%0.1359%0.1359%3.7748%4.4545%11.4505%
Two Pairs2.0262%1.9548%1.8835%1.8121%1.7407%13.3691%22.7864%34.2369%
Single Pair2.123%2.1409%2.1409%2.123%2.1052%35.3652%45.9982%80.2351%
High Card9.2516%5.6071%3.0372%1.3083%0.4673%0.0935%19.7649%100%

Total Possibilities


The main point of interest in this scenario is:
1. The probability of getting anything good is small.

You may now like to continue to the after the flop or after the turn scenarios:
After the Flop Scenarios
After the Turn Scenarios

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