Estimate Probabilities of Card Games

A practical example of how you can calculate Card Probabilities with Monte Carlo Simulation and Numerically

George Pipis
Aug 25, 2020 · 5 min read
Photo by Amanda Jones on Unsplash

We are going to show how we can estimate card probabilities by applying Monte Carlo Simulation and how we can solve them numerically in Python. The first thing that we need to do is to create a deck of 52 cards.

How to Generate a Deck of Cards

import itertools, random

And we get:

[('A', 'Spade'),
('A', 'Heart'),
('A', 'Diamond'),
('A', 'Club'),
('2', 'Spade'),
('2', 'Heart'),
('2', 'Diamond'),
('2', 'Club'),
('3', 'Spade'),
('3', 'Heart'),
('3', 'Diamond'),
('3', 'Club'),
('4', 'Spade'),
('4', 'Heart'),
('4', 'Diamond'),
('4', 'Club'),
('5', 'Spade'),
('5', 'Heart'),
...

How to Shuffle the Deck

# shuffle the cards
random.shuffle(deck)
deck[0:10]

And we get:

[('6', 'Club'),
('8', 'Spade'),
('J', 'Heart'),
('10', 'Heart'),
('Q', 'Spade'),
('7', 'Diamond'),
('K', 'Diamond'),
('J', 'Club'),
('J', 'Diamond'),
('A', 'Club')]

How to Sort the Deck

For some probabilities where the order does not matter, a good trick is to sort the cards. The following commands can be helpful.

# sort the deck
sorted(deck)

How to Remove Cards from the Deck

Depending on the Games and the problem that we need to solve, sometimes there is a need to remove from the Deck the cards which have already been served. The commands that we can use are the following:

# assume that card is a tuple like ('J', 'Diamond')
deck.remove(card)

Part 1: Estimate Card Probabilities with Monte Carlo Simulation

Question 1: What is the probability that when two cards are drawn from a deck of cards without a replacement that both of them will be Ace?

Let’s say that we were not familiar with formulas or that the problem was more complicated. We could find an approximate probability with a Monte Carlo Simulation (10M Simulations)

N = 10000000
double_aces = 0

And we get 0.0045214 where the actual probability is 0.0045

Question 2: What is the probability of two Aces in 5 card hand without replacement.

FYI: In case you want to solve it explicitly by applying mathematical formulas:

from scipy.stats import hypergeom

And we get 0.03992981808107859. Let’s try to solve it by applying simulation:

N = 10000000
double_aces = 0

And we get 0.0398805. Again, quite close to the actual probability

Question 3: What is the probability of being dealt a flush (5 cards of all the same suit) from the first 5 cards in a deck?

If you would like to see the mathematical solution of this question you can visit PredictiveHacks. Let’s solve it again by applying a Monte Carlo Simulation.

N = 10000000

And we get 0.0019823 which is quite close to the actual probability which is 0.00198.

Question 4: What is the probability of being dealt a royal flush from the first 5 cards in a deck?

The actual probability of this case is around 0.00000154. Let’s see if the Monte Carlo works with such small numbers.

# royal flush

And we get 1.5e-06 which is a very good estimation.

Part 2: Calculate Exact Card Probabilities Numerically

Above, we showed how we can calculate the Card Probabilities explicitly by applying mathematical formulas and how we can estimate them by applying Monte Carlo Simulation. Now, we will show how we can get the exact probability using Python. This is not always applicable but let’s try to solve the questions of Part 1.

The logic here is to generate all the possible combinations and then to calculate the ratio. Let’s see how we can get all the possible 5-hand cards of a 52-card deck.

# importing modules

And we get 2598960 as expected.

Question 1: What is the probability that when two cards are drawn from a deck of cards without a replacement that both of them will be Ace?

# importing modules

And we get 0.004524886877828055.

Question 2: What is the probability of two Aces in 5 card hand without replacement.

# get the (52 5) combinations

And we get 0.03992981808107859

Question 3: What is the probability of being dealt a flush (5 cards of all the same suit) from the first 5 cards in a deck?

# get the (52 5) combinations
all_possible_by_5_combinations = list(combinations(deck,5))

And we get 0.0019807923169267707

Question 4: What is the probability of being dealt a royal flush from the first 5 cards in a deck?

# get the (52 5) combinations

And we get 1.5390771693292702e-06

Discussion

Computing Power enables us to estimate and even calculate complicated probabilities without being necessary to be experts in Statistics and Probabilities. In this post, we provided some relatively simple examples. The same logic can be extended to more advanced and complicated problems in many fields, like simulating card games, lotteries, gambling games etc.

Originally published at https://predictivehacks.com.

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