This is my first post. It is 50% an anecdote about an experience that inspired me to start this blog, and 50% a showcase of how a domain can be modelled using a few lines of Python.
I was recently staying at a hotel situated above a casino. Curiosity got the better of me on the last day and I decided to write off what remained of my travel money. Being a gambling novice, I avoided the tables and made a beeline for the cluster of noisy and garish machines. Among various themed slot games, I found something vaguely familiar: Jacks-or-better video poker.
The premise was simple: the player is dealt five cards and can hold onto any of these, the remainder being replaced by new cards. They win money if they get a pair of jacks or higher, or any other poker hand.
Is the machine lying to me?
Despite getting an occasional winning hand, I steadily lost money. I began to suspect this was partly due to the machine hinting that I should hold the wrong cards. Going on my statistical intuition I began ignoring suggestions to hold certain cards.
Thinking on my feet
To me, the most glaringly bad suggestion was to hold a pair of anything from twos to tens. On their own merits, these would be worthless. I was being encouraged to bet on three of a kind or something better.
The probability of getting the same value card for one of the three new cards would be something like 2/50 + 2/49 + 2/48. Let's say 6/50. While holding a pair precludes straights and flushes, there is also a higher-than-normal chance of a full house and four of a kind. Despite this, I was sure that getting a completely new hand would give me a better chance of winning.
Doing my homework
When I got home I decided to crunch some numbers to put my intuition to the test.
Let's say I opted to hold a pair of twos: a two of hearts and a two of diamonds.
>>> SUITS = 'CDHS'
>>> VALUES = '23456789TJQKA'
>>> full_deck = [val + suit for suit in SUITS for val in VALUES]
>>> my_deck = full_deck.copy()
>>> hold = []
>>> hold.append(my_deck.pop(my_deck.index("2H")))
>>> hold.append(my_deck.pop(my_deck.index("2D")))
Bear with me. For a fair comparison, I need to compute all possible hands for two cases:
if I hold the pair of twos
if I don't hold anything
Python's itertools package comes in handy here.
>>> import itertools
>>> threes = list(itertools.combinations(my_deck, 3))
>>> possible_hands_if_i_hold = [[*three, *hold] for three in threes]
>>> len(possible_hands_if_i_hold)
22100
>>> all_possible_hands = list(itertools.combinations(full_deck, 5))
>>> len(all_possible_hands)
2598960
I didn't expect orders of magnitude fewer possibilities when holding, but it makes sense.
Here's the function I devised to determine if a poker hand wins.
def rank_hand(hand):
if poker.is_royal_flush(hand):
return Hands.ROYAL_FLUSH.value
elif poker.is_straight_flush(hand):
return Hands.STRAIGHT_FLUSH.value
elif poker.is_four_of_a_kind(hand):
return Hands.FOUR_OF_KIND.value
elif poker.is_full_house(hand):
return Hands.FULL_HOUSE.value
elif poker.is_flush(hand):
return Hands.FLUSH.value
elif poker.is_straight(hand):
return Hands.STRAIGHT.value
elif poker.is_three_of_a_kind(hand):
return Hands.THREE_OF_A_KIND.value
elif poker.is_two_pair(hand):
return Hands.TWO_PAIR.value
elif is_pair(hand):
return Hands.PAIR_JACKS_OR_BETTER.value
return 0
This function relies on predicates like this one:
def is_pair(hand):
"""Jacks-or-better variant of is_pair"""
hand_values, _ = zip(*hand)
hand_ranks = sorted(poker.VALUES.find(val) for val in hand_values)
rank_counter = collections.Counter(hand_ranks)
for rank, count in rank_counter.items():
if count == 2 and rank >= poker.VALUES.find("J"):
return True
return False
You can find the rest of the predicates here.
Because Python integers can be truthy (and for that matter booleans are integers e.g. True + True == 2), rank_hand also functions as a predicate and can be used with filter to get a winning subset of all hands.
>>> import poker, jacks_or_better
>>> len(list(filter(jacks_or_better.rank_hand, all_possible_hands)))/ len(all_possible_hands)
0.20588235294117646
>>> len(list(filter(jacks_or_better.rank_hand, possible_hands_if_i_hold)))/ len(possible_hands_if_i_hold)
0.2816326530612245
Numbers don't lie! It turns out that I was throwing away an 8% chance of winning something by not holding onto low pairs.
Out of curiosity, what winning hands did I have with a pair of twos?
>>> ranks_if_hold = collections.Counter(sorted(map(jacks_or_better.rank_hand, possible_hands_if_i_hold)))
>>> hands_by_rank = {hand.value: hand.name for hand in jacks_or_better.Hands}
>>> for rank, count in ranks_if_hold.items(): print(hands_by_rank.get(rank), count)
None 14080
FOUR_OF_KIND 48
FULL_HOUSE 192
THREE_OF_A_KIND 2112
TWO_PAIR 3168
These aren't the best hands, but as those are rare, these are still probably the best bet.
Although gambling machines are designed to give the house at least a marginal edge on players, I wonder if I could have won at video poker with an optimal strategy.
Conclusions
There are a couple of things I can take from this:
I made more money playing an Egyptian-themed slot game than poker, although all I did was repeatedly press a button like a rat in some kind of kaleidoscopic Skinner box. I still prefer video poker as it leaves some room for the player's agency.
My intuitive grasp of statistics is shaky. I'll do some research before setting foot in a casino again.
Python is a great tool for quickly modelling a problem domain. However, my laptop takes a few seconds to rank all possible hands. Rust's speed and expressive type system might make it a better fit for further work with modelling Poker. For example, I could enumerate and classify different starting hands and factor in the payout of winning hands.
I hope you enjoyed this little post.
I'll follow up soon with more substantial writing on backend software development and cloud engineering.