# Ace-King Hand in Poker – How to Play It, Odds & More

AKo an AKs is a quality hand in poker.

It is a slight underdog against all pairs, but it holds a lot of equity.

In this article, we look at the possibilities with this hand.

In Texas Hold ‘Em, when holding Ace-King offsuit (AKo), the odds of hitting a straight or a flush by the river (after all five community cards are dealt) can be calculated as follows:

1. Odds of Hitting a Straight: To hit a straight with AKo, you would typically need three of the five community cards to form a sequence with your AK. For example, if you hold A♠ K♦, you need the community cards to include something like Q-J-T or T-Q-J (in any suit combination) to complete a straight. The specific odds can be quite complex to calculate, as they depend on the exact cards that come on the flop, turn, and river.
2. Odds of Hitting a Flush: Since AKo is offsuit, hitting a flush is not possible unless you get four community cards of the same suit as either your Ace or King. This is highly unlikely, as it requires a very specific set of community cards.

The precise odds for these scenarios involve detailed probability calculations and can vary depending on the exact community cards dealt in each stage (flop, turn, river). However, it’s generally understood that hitting a straight or a flush with AKo is not very common, especially for a flush, given the offsuit nature of the hand.

For a more exact calculation, one would need to run a simulation or use advanced poker probability software that takes into account all possible combinations of community cards.

## Ace-King Offsuit Odds of Hitting a Straight or Flush

For a hand of Ace of Hearts and King of Diamonds in Texas Hold ‘Em, the odds of hitting a straight or a flush by the river are as follows:

1. Odds of Hitting a Straight: Approximately 2.96%
2. Odds of Hitting a Flush: Approximately 1.97%

These calculations consider all possible combinations of the five community cards given the initial hand.

Keep in mind that these probabilities are theoretical and assume a standard deck with no additional information about the cards already played or the hands of other players.

Here is some Python code on how we came up with this:

```# Constants
SUITS = ['Hearts', 'Diamonds', 'Clubs', 'Spades']
RANKS = ['2', '3', '4', '5', '6', '7', '8', '9', 'T', 'J', 'Q', 'K', 'A']
DECK = [rank + suit[0] for suit in SUITS for rank in RANKS]

# Corrected function to calculate odds
def calculate_odds_corrected(hand):
# Remove the hand cards from the deck
remaining_deck = DECK.copy()
for card in hand:
remaining_deck.remove(card)

# All possible combinations of 5 cards from the remaining deck
community_combinations = list(combinations(remaining_deck, 5))

# Counters for straight and flush
straight_count = 0
flush_count = 0

# Check each combination
for community in community_combinations:
combined_hand = hand + list(community)
if is_straight(combined_hand):
straight_count += 1
if is_flush(combined_hand):
flush_count += 1

total_combinations = len(community_combinations)
return straight_count / total_combinations, flush_count / total_combinations

# Recalculating with the corrected hand and function
hand_corrected = ['AH', 'KD'] # Ace of Hearts, King of Diamonds
straight_odds_corrected, flush_odds_corrected = calculate_odds_corrected(hand_corrected)
straight_odds_corrected, flush_odds_corrected

```

## Odds of a Flush with Ace-King Suited

Let’s say we’re dealt the Ace-King of hearts.

When holding two hearts in your hand pre-flop in Texas Hold ‘Em, the odds of hitting a flush (five cards of hearts) by the river can be calculated as follows:

1. Deck Composition: Initially, there are 13 hearts in a standard 52-card deck. If you have two of them in your hand, there are 11 hearts left in the 50 remaining cards.
2. Calculating the Odds: You need at least three of the five community cards to be hearts to complete your flush. This calculation involves finding the probability of getting 3, 4, or all 5 hearts in the community cards.

Let’s calculate these probabilities.

The probability of hitting a flush by the river when holding two hearts in your hand pre-flop in Texas Hold ‘Em is approximately 6.40%.

This calculation considers all scenarios where you could get at least three more hearts among the five community cards.

This is a theoretical probability and assumes a standard deck without considering other players’ hands or any specific game dynamics.

```from math import comb

# Constants
total_remaining_cards = 50 # Total cards remaining after dealing two to the player
hearts_remaining = 11 # Hearts remaining after the player has two

# Function to calculate the probability of hitting a flush
def flush_probability(hearts_needed):
# Number of ways to draw 'hearts_needed' hearts and the rest non-hearts
ways_to_draw_hearts = comb(hearts_remaining, hearts_needed)
ways_to_draw_non_hearts = comb(total_remaining_cards - hearts_remaining, 5 - hearts_needed)

# Total ways to draw 5 community cards
total_ways_to_draw_5 = comb(total_remaining_cards, 5)

# Probability of this specific combination
return ways_to_draw_hearts * ways_to_draw_non_hearts / total_ways_to_draw_5

# Calculating the probabilities
prob_3_hearts = flush_probability(3)
prob_4_hearts = flush_probability(4)
prob_5_hearts = flush_probability(5)

# Total probability of hitting a flush
total_flush_probability = prob_3_hearts + prob_4_hearts + prob_5_hearts
total_flush_probability```

## What are the odds of hitting an overpair with AKs or AKo?

To calculate the odds of hitting an overpair with Ace-King suited (AKs) or Ace-King offsuit (AKo) in Texas Hold ‘Em, you’re essentially calculating the probability of at least one Ace or King appearing in the five community cards. The calculation is the same for both AKs and AKo since the suit doesn’t impact the likelihood of hitting an overpair in this scenario.

Here’s how to calculate it:

1. Deck Composition: Initially, there are 4 Aces and 4 Kings in a standard 52-card deck. If you have an Ace and a King, there are 3 Aces and 3 Kings left in the 50 remaining cards (since your hand’s cards are removed from the deck).
2. Calculating the Odds: You want to find the probability that at least one of the five community cards is an Ace or a King. This can be calculated directly, or more simply, by calculating the probability of not hitting an Ace or a King and subtracting this from 1.

Let’s calculate these probabilities.

The probability of hitting an overpair with Ace-King suited (AKs) or Ace-King offsuit (AKo) in Texas Hold ‘Em is approximately 47.23%. This means you have nearly a 50% chance of at least one Ace or King appearing in the five community cards, thus giving you an overpair.

```# Constants for the calculation
total_cards_remaining = 50 # Total cards remaining after the hand is dealt
aces_and_kings_remaining = 3 + 3 # 3 Aces and 3 Kings remaining

# Probability of not drawing an Ace or a King in each community card
prob_not_ace_or_king_per_card = (total_cards_remaining - aces_and_kings_remaining) / total_cards_remaining

# Probability of not drawing an Ace or a King in all 5 community cards
prob_not_ace_or_king_all = prob_not_ace_or_king_per_card ** 5

# Probability of hitting at least one Ace or King (thus making an overpair)
prob_hitting_overpair = 1 - prob_not_ace_or_king_all
prob_hitting_overpair```