• Focus: Statistics of starting hands in Texas Hold’em
• Key questions:
  • How many starting hands are possible?
  • How rare are pocket pairs?
  • What is the probability of suited vs offsuit hands?
• Tools:
  • Combinatorics
  • ggplot2
  • plotly (3D)

In Texas Hold’em, you are dealt two cards from a 52-card deck,
and order does not matter.

\[ \text{Total Hands} = \binom{52}{2} \]

\[ \binom{52}{2} = \frac{52 \times 51}{2} = 1326 \]

There are 1326 distinct starting hands in Texas Hold’em.

A pocket pair means both cards share the same rank (e.g., AA, KK).

There are 4 cards of each rank.
Number of ways to get a specific pocket pair (like AA):

\[ \binom{4}{2} = 6 \]

Probability of a specific pocket pair (e.g., AA):

\[ P(\text{AA}) = \frac{6}{1326} \approx 0.0045 \]

Probability of any pocket pair (13 different ranks):

\[ P(\text{any pair}) = \frac{13 \times 6}{1326} \approx 0.058 \]

# Total possible unordered starting hands 
total_hands <- choose(52, 2)

# Specific pocket pair (e.g. two A's)
specific_pair_combos <- choose(4, 2)
p_specific_pair <- specific_pair_combos / total_hands

# Any pocket pair
all_pair_combos <- 13 * specific_pair_combos
p_any_pair <- all_pair_combos / total_hands

# Suited non-pair combinations
suited_combos <- 4 * choose(13, 2)
p_any_suited <- suited_combos / total_hands

# Offsuit non-pair combinations
nonpair_combos <- total_hands - all_pair_combos
offsuit_combos <- nonpair_combos - suited_combos
p_any_offsuit <- offsuit_combos / total_hands

c(
  AA_probability       = p_specific_pair,
  any_pair_probability = p_any_pair,
  suited_probability   = p_any_suited,
  offsuit_probability  = p_any_offsuit
)

##       AA_probability any_pair_probability   suited_probability 
##          0.004524887          0.058823529          0.235294118 
##  offsuit_probability 
##          0.705882353

premium_hands <- data.frame(
  # AKs = Ace-King of the same suit
  Hand   = c("AA", "KK", "QQ", "JJ", "AKs"),
  # 6 combos per pocket pair, 4 combos for AK suited
  Combos = c(6, 6, 6, 6, 4)   
)

premium_hands$Probability <- premium_hands$Combos / total_hands

premium_hands

##   Hand Combos Probability
## 1   AA      6 0.004524887
## 2   KK      6 0.004524887
## 3   QQ      6 0.004524887
## 4   JJ      6 0.004524887
## 5  AKs      4 0.003016591