Probability of Drawing a Full House

I recently saw someone mention that they received an interview question for a DS position in which they were asked to calculate the probability of drawing a full house when drawing 5 cards from a standard 52-card deck.

So let’s solve that in Julia.

Solving analytically

The function we want is binomial(n::Integer, k::Integer), which returns the binomial coefficient – the number of ways to choose k out of n items.

Let’s look at some examples. First, if we try 4C1 (4 choose 1), we expect to just get 4 – there are 4 different ways to choose 1 item from a group of 4 items.

binomial(4, 1)

4

Now imagine we choose 2 different items from a group of 4. We expect to get 6 (assuming we don’t care about order, i.e. that 1,2 is the same as 2,1):

1. 1, 2
2. 1, 3
3. 1, 4
4. 2, 3
5. 2, 4
6. 3, 4

binomial(4, 2)

6

So let’s solve the actual problem now. A full house is 5 cards comprising 3-of-a-kind and a pair. There are 52 cards in a deck – 4 suits comprising 13 unique values (2, 3, …, Ace) each.

The approach here is to calculate the number of ways to get a full house and divide that by the number of ways to draw 5 cards from a deck. We can start with the number of ways to draw 5 cards from a deck (the denominator) first, since it’s the most straightforward:

denom = binomial(52, 5)

2598960

Then let’s calculate the number of ways we can get three of a kind. There are 13 different card values and 4 different suits. We need to choose 1 value with 3 different suits:

• binomial(13, 1) gives us the number of ways to choose 1 value from 13 options (which is just 13)
• binomial(4, 3) gives us the number of ways to choose 3 different suits from 4 possible options

And then since this is probability, we multiply everything together:

three_kind = binomial(13, 1) * binomial(4, 3)

52

Then we do the same thing for drawing a pair. There are now 12 different card values (we can’t get a pair of the value that we already drew three-of-a-kind for), and we need to choose 1 value with 2 different suits:

two_kind = binomial(12, 1) * binomial(4, 2)

72

And from here, we can estimate the probability of a full house by multiplying and dividing:

(three_kind * two_kind) / denom

0.0014405762304921968

So there’s a 0.144% chance of drawing a full house from a typical 52-card deck.

Solving with simulation

We could also take a simulation-approach to solving this. First, let’s create a deck of cards.

using Random
using StatsBase

Random.seed!(0408)

deck = collect(repeat(1:13, 4))

52-element Vector{Int64}:
  1
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13
  ⋮
  2
  3
  4
  5
  6
  7
  8
  9
 10
 11
 12
 13

Then we’ll create a few functions to help us with the simulation:

1. make_hands() will draw n 5-card hands from the deck;
2. is_full_house() will check whether any given hand is a full house;
3. count_full_house() takes a vector of hands and counts the number of them that have a full house

function make_hands(deck::AbstractVector{<:Integer}, n::Int64)
    v = Vector{Vector{Int64}}(undef, n)
    for i in 1:n
        v[i] = sample(deck, 5; replace=false)
    end
    return v
end

function is_full_house(hand::AbstractVector{<:Integer})
    return extrema(values(countmap(hand))) == (2, 3)
end

function count_full_house(hands::Vector{Vector{Int64}})
    s = 0
    for i in eachindex(hands)
        if is_full_house(hands[i])
            s += 1
        end
    end
    return s
end

count_full_house (generic function with 1 method)

Then from here we just run our simulation.

n = 1_000_000

hands = make_hands(deck, n);

count_full_house(hands) / n

0.001446

And we see that we get roughly the same answer as we did previously.

Reuse

Citation

BibTeX citation:

@online{ekholm2024,
  author = {Ekholm, Eric},
  title = {Probability of {Drawing} a {Full} {House}},
  date = {2024-03-29},
  url = {https://www.ericekholm.com/posts/full-house},
  langid = {en}
}

For attribution, please cite this work as:

Ekholm, Eric. 2024. “Probability of Drawing a Full House.” March 29, 2024. https://www.ericekholm.com/posts/full-house.