https://thefiddler.substack.com/p/can-you-throw-the-card-game

throw_card_game.jl

### A Pluto.jl notebook ###
# v0.19.29

using Markdown
using InteractiveUtils

# ╔═╡ dff33215-c1f4-47db-bfd1-24de63c9f67d
using Distributions

# ╔═╡ 5d67bf6a-6a9d-11ee-3858-5be18c658ac6
md"""
# [Throw Card Game](https://thefiddler.substack.com/i/137914922/this-weeks-fiddler)

Every so often, I play the card game Slap with someone who really doesn’t like to lose. (You know who you are.) But before we get into all that, here are the rules of the game:

- Two players are both dealt 26 cards from a standard deck.
- From there, they take turns putting down one card at a time. Let’s interchangeably designate the players as “A” and “B.”
- If A puts down a jack, then B puts down one card. If it’s a face card, the game continues with A. Otherwise, A wins the round and adds those cards to the bottom of their pile so that the last card played is at the very bottom of the pile.
- If A puts down a queen, B has two chances to put down a face card, at which point the game would continue with A. Otherwise, A wins the round and collects the cards.
- If A puts down a king, B has three chances to put down a face card, at which point the game would continue with A. Otherwise, A wins the round and collects the cards.
- If A puts down an ace, B has four chances to put down a face card, at which point the game would continue with A. Otherwise, A wins the round and collects the cards.
- If, at any point, two cards have been played in succession (by different players or the same player) that have the same number or face, then both players can slap the pile of played cards to win the round and collect the cards.
- Whoever won the previous round plays the first card in the next round.

Now look. I really like this opponent and I don’t want to hurt their feelings. And so, while shuffling, I rig the deck so that they get a majority of the jacks, a majority of the queens, a majority of the kings, and a majority of the aces.

Nobody likes a cheater, and so my opponent rightfully accuses me of rigging the game. In my defense, I say that the probability of one player randomly receiving the majority of each kind of face card is reasonable. What is this probability?
"""

# ╔═╡ 2b227cb3-860a-464f-bba5-220fec035d12
md"""
## Answer

Probability of getting a majority of each type of face card. There are three types of face cards:
- Jack
- Queen
- King
And four of each type. A majority of each type means getting three or four. Assuming random distribution, the probability of getting a majority of a single type is a [binomial distribution](https://en.wikipedia.org/wiki/Binomial_distribution):

The PMF is 

$pmf(n, k, p) = \binom{n}{k} p^k (1-p)^{n-k}$
"""

# ╔═╡ 81bb0432-587e-4921-8da8-a3fcfbb8325d
"""
The pmf for getting \$k\$ face cards, assuming \$p = 0.5\$
"""
face_card_pmf(k) = pdf(Binomial(4, 0.5), k)

# ╔═╡ 6f2b0ba5-e30f-4e63-a42b-b9614b41d897
face_card_pmf.(0:4)

# ╔═╡ 929a5549-0e8a-4df3-b378-b74cef110bd0
md"""
So the probability of getting 3 or 4 is:
"""

# ╔═╡ 31f3abea-6d66-488c-a9b6-6b3fdb733447
p_3_or_4 = sum(face_card_pmf.([3, 4]))

# ╔═╡ 34e69d0c-980f-4d45-bbbf-6241a9061be5
md"""
And the probability of getting a majority three times in a row is that to the power 3.
"""

# ╔═╡ 20d3ddda-d10e-42d2-b05b-b8af1b9cad5e
p_majority_all_face_cards = p_3_or_4^3

# ╔═╡ 36957cfc-a427-43b1-9999-7fedc584e789
md"""
So `p_majority_all_face_cards = 0.030517578125`
"""

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f"

[compat]
Distributions = "~0.25.102"
"""

# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
# This file is machine-generated - editing it directly is not advised

julia_version = "1.9.3"
manifest_format = "2.0"
project_hash = "b8fff1bcbf899ee7a8daed9608ac64571247262f"

[[deps.ArgTools]]
uuid = "0dad84c5-d112-42e6-8d28-ef12dabb789f"
version = "1.1.1"

[[deps.Artifacts]]
uuid = "56f22d72-fd6d-98f1-02f0-08ddc0907c33"

[[deps.Base64]]
uuid = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f"

[[deps.Calculus]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "f641eb0a4f00c343bbc32346e1217b86f3ce9dad"
uuid = "49dc2e85-a5d0-5ad3-a950-438e2897f1b9"
version = "0.5.1"

[[deps.Compat]]
deps = ["UUIDs"]
git-tree-sha1 = "8a62af3e248a8c4bad6b32cbbe663ae02275e32c"
uuid = "34da2185-b29b-5c13-b0c7-acf172513d20"
version = "4.10.0"
weakdeps = ["Dates", "LinearAlgebra"]

    [deps.Compat.extensions]
    CompatLinearAlgebraExt = "LinearAlgebra"

[[deps.CompilerSupportLibraries_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "e66e0078-7015-5450-92f7-15fbd957f2ae"
version = "1.0.5+0"

[[deps.DataAPI]]
git-tree-sha1 = "8da84edb865b0b5b0100c0666a9bc9a0b71c553c"
uuid = "9a962f9c-6df0-11e9-0e5d-c546b8b5ee8a"
version = "1.15.0"

[[deps.DataStructures]]
deps = ["Compat", "InteractiveUtils", "OrderedCollections"]
git-tree-sha1 = "3dbd312d370723b6bb43ba9d02fc36abade4518d"
uuid = "864edb3b-99cc-5e75-8d2d-829cb0a9cfe8"
version = "0.18.15"

[[deps.Dates]]
deps = ["Printf"]
uuid = "ade2ca70-3891-5945-98fb-dc099432e06a"

[[deps.Distributions]]
deps = ["FillArrays", "LinearAlgebra", "PDMats", "Printf", "QuadGK", "Random", "SpecialFunctions", "Statistics", "StatsAPI", "StatsBase", "StatsFuns", "Test"]
git-tree-sha1 = "3d5873f811f582873bb9871fc9c451784d5dc8c7"
uuid = "31c24e10-a181-5473-b8eb-7969acd0382f"
version = "0.25.102"

    [deps.Distributions.extensions]
    DistributionsChainRulesCoreExt = "ChainRulesCore"
    DistributionsDensityInterfaceExt = "DensityInterface"

    [deps.Distributions.weakdeps]
    ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
    DensityInterface = "b429d917-457f-4dbc-8f4c-0cc954292b1d"

[[deps.DocStringExtensions]]
deps = ["LibGit2"]
git-tree-sha1 = "2fb1e02f2b635d0845df5d7c167fec4dd739b00d"
uuid = "ffbed154-4ef7-542d-bbb7-c09d3a79fcae"
version = "0.9.3"

[[deps.Downloads]]
deps = ["ArgTools", "FileWatching", "LibCURL", "NetworkOptions"]
uuid = "f43a241f-c20a-4ad4-852c-f6b1247861c6"
version = "1.6.0"

[[deps.DualNumbers]]
deps = ["Calculus", "NaNMath", "SpecialFunctions"]
git-tree-sha1 = "5837a837389fccf076445fce071c8ddaea35a566"
uuid = "fa6b7ba4-c1ee-5f82-b5fc-ecf0adba8f74"
version = "0.6.8"

[[deps.FileWatching]]
uuid = "7b1f6079-737a-58dc-b8bc-7a2ca5c1b5ee"

[[deps.FillArrays]]
deps = ["LinearAlgebra", "Random"]
git-tree-sha1 = "a20eaa3ad64254c61eeb5f230d9306e937405434"
uuid = "1a297f60-69ca-5386-bcde-b61e274b549b"
version = "1.6.1"
weakdeps = ["SparseArrays", "Statistics"]

    [deps.FillArrays.extensions]
    FillArraysSparseArraysExt = "SparseArrays"
    FillArraysStatisticsExt = "Statistics"

[[deps.HypergeometricFunctions]]
deps = ["DualNumbers", "LinearAlgebra", "OpenLibm_jll", "SpecialFunctions"]
git-tree-sha1 = "f218fe3736ddf977e0e772bc9a586b2383da2685"
uuid = "34004b35-14d8-5ef3-9330-4cdb6864b03a"
version = "0.3.23"

[[deps.InteractiveUtils]]
deps = ["Markdown"]
uuid = "b77e0a4c-d291-57a0-90e8-8db25a27a240"

[[deps.IrrationalConstants]]
git-tree-sha1 = "630b497eafcc20001bba38a4651b327dcfc491d2"
uuid = "92d709cd-6900-40b7-9082-c6be49f344b6"
version = "0.2.2"

[[deps.JLLWrappers]]
deps = ["Artifacts", "Preferences"]
git-tree-sha1 = "7e5d6779a1e09a36db2a7b6cff50942a0a7d0fca"
uuid = "692b3bcd-3c85-4b1f-b108-f13ce0eb3210"
version = "1.5.0"

[[deps.LibCURL]]
deps = ["LibCURL_jll", "MozillaCACerts_jll"]
uuid = "b27032c2-a3e7-50c8-80cd-2d36dbcbfd21"
version = "0.6.3"

[[deps.LibCURL_jll]]
deps = ["Artifacts", "LibSSH2_jll", "Libdl", "MbedTLS_jll", "Zlib_jll", "nghttp2_jll"]
uuid = "deac9b47-8bc7-5906-a0fe-35ac56dc84c0"
version = "7.84.0+0"

[[deps.LibGit2]]
deps = ["Base64", "NetworkOptions", "Printf", "SHA"]
uuid = "76f85450-5226-5b5a-8eaa-529ad045b433"

[[deps.LibSSH2_jll]]
deps = ["Artifacts", "Libdl", "MbedTLS_jll"]
uuid = "29816b5a-b9ab-546f-933c-edad1886dfa8"
version = "1.10.2+0"

[[deps.Libdl]]
uuid = "8f399da3-3557-5675-b5ff-fb832c97cbdb"

[[deps.LinearAlgebra]]
deps = ["Libdl", "OpenBLAS_jll", "libblastrampoline_jll"]
uuid = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"

[[deps.LogExpFunctions]]
deps = ["DocStringExtensions", "IrrationalConstants", "LinearAlgebra"]
git-tree-sha1 = "7d6dd4e9212aebaeed356de34ccf262a3cd415aa"
uuid = "2ab3a3ac-af41-5b50-aa03-7779005ae688"
version = "0.3.26"

    [deps.LogExpFunctions.extensions]
    LogExpFunctionsChainRulesCoreExt = "ChainRulesCore"
    LogExpFunctionsChangesOfVariablesExt = "ChangesOfVariables"
    LogExpFunctionsInverseFunctionsExt = "InverseFunctions"

    [deps.LogExpFunctions.weakdeps]
    ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
    ChangesOfVariables = "9e997f8a-9a97-42d5-a9f1-ce6bfc15e2c0"
    InverseFunctions = "3587e190-3f89-42d0-90ee-14403ec27112"

[[deps.Logging]]
uuid = "56ddb016-857b-54e1-b83d-db4d58db5568"

[[deps.Markdown]]
deps = ["Base64"]
uuid = "d6f4376e-aef5-505a-96c1-9c027394607a"

[[deps.MbedTLS_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "c8ffd9c3-330d-5841-b78e-0817d7145fa1"
version = "2.28.2+0"

[[deps.Missings]]
deps = ["DataAPI"]
git-tree-sha1 = "f66bdc5de519e8f8ae43bdc598782d35a25b1272"
uuid = "e1d29d7a-bbdc-5cf2-9ac0-f12de2c33e28"
version = "1.1.0"

[[deps.MozillaCACerts_jll]]
uuid = "14a3606d-f60d-562e-9121-12d972cd8159"
version = "2022.10.11"

[[deps.NaNMath]]
deps = ["OpenLibm_jll"]
git-tree-sha1 = "0877504529a3e5c3343c6f8b4c0381e57e4387e4"
uuid = "77ba4419-2d1f-58cd-9bb1-8ffee604a2e3"
version = "1.0.2"

[[deps.NetworkOptions]]
uuid = "ca575930-c2e3-43a9-ace4-1e988b2c1908"
version = "1.2.0"

[[deps.OpenBLAS_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "Libdl"]
uuid = "4536629a-c528-5b80-bd46-f80d51c5b363"
version = "0.3.21+4"

[[deps.OpenLibm_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "05823500-19ac-5b8b-9628-191a04bc5112"
version = "0.8.1+0"

[[deps.OpenSpecFun_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "13652491f6856acfd2db29360e1bbcd4565d04f1"
uuid = "efe28fd5-8261-553b-a9e1-b2916fc3738e"
version = "0.5.5+0"

[[deps.OrderedCollections]]
git-tree-sha1 = "2e73fe17cac3c62ad1aebe70d44c963c3cfdc3e3"
uuid = "bac558e1-5e72-5ebc-8fee-abe8a469f55d"
version = "1.6.2"

[[deps.PDMats]]
deps = ["LinearAlgebra", "SparseArrays", "SuiteSparse"]
git-tree-sha1 = "3d8af7d689bd228a3a6c2298a4e6d5f5944accb8"
uuid = "90014a1f-27ba-587c-ab20-58faa44d9150"
version = "0.11.26"

[[deps.Pkg]]
deps = ["Artifacts", "Dates", "Downloads", "FileWatching", "LibGit2", "Libdl", "Logging", "Markdown", "Printf", "REPL", "Random", "SHA", "Serialization", "TOML", "Tar", "UUIDs", "p7zip_jll"]
uuid = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f"
version = "1.9.2"

[[deps.Preferences]]
deps = ["TOML"]
git-tree-sha1 = "00805cd429dcb4870060ff49ef443486c262e38e"
uuid = "21216c6a-2e73-6563-6e65-726566657250"
version = "1.4.1"

[[deps.Printf]]
deps = ["Unicode"]
uuid = "de0858da-6303-5e67-8744-51eddeeeb8d7"

[[deps.QuadGK]]
deps = ["DataStructures", "LinearAlgebra"]
git-tree-sha1 = "9ebcd48c498668c7fa0e97a9cae873fbee7bfee1"
uuid = "1fd47b50-473d-5c70-9696-f719f8f3bcdc"
version = "2.9.1"

[[deps.REPL]]
deps = ["InteractiveUtils", "Markdown", "Sockets", "Unicode"]
uuid = "3fa0cd96-eef1-5676-8a61-b3b8758bbffb"

[[deps.Random]]
deps = ["SHA", "Serialization"]
uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"

[[deps.Reexport]]
git-tree-sha1 = "45e428421666073eab6f2da5c9d310d99bb12f9b"
uuid = "189a3867-3050-52da-a836-e630ba90ab69"
version = "1.2.2"

[[deps.Rmath]]
deps = ["Random", "Rmath_jll"]
git-tree-sha1 = "f65dcb5fa46aee0cf9ed6274ccbd597adc49aa7b"
uuid = "79098fc4-a85e-5d69-aa6a-4863f24498fa"
version = "0.7.1"

[[deps.Rmath_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6ed52fdd3382cf21947b15e8870ac0ddbff736da"
uuid = "f50d1b31-88e8-58de-be2c-1cc44531875f"
version = "0.4.0+0"

[[deps.SHA]]
uuid = "ea8e919c-243c-51af-8825-aaa63cd721ce"
version = "0.7.0"

[[deps.Serialization]]
uuid = "9e88b42a-f829-5b0c-bbe9-9e923198166b"

[[deps.Sockets]]
uuid = "6462fe0b-24de-5631-8697-dd941f90decc"

[[deps.SortingAlgorithms]]
deps = ["DataStructures"]
git-tree-sha1 = "c60ec5c62180f27efea3ba2908480f8055e17cee"
uuid = "a2af1166-a08f-5f64-846c-94a0d3cef48c"
version = "1.1.1"

[[deps.SparseArrays]]
deps = ["Libdl", "LinearAlgebra", "Random", "Serialization", "SuiteSparse_jll"]
uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"

[[deps.SpecialFunctions]]
deps = ["IrrationalConstants", "LogExpFunctions", "OpenLibm_jll", "OpenSpecFun_jll"]
git-tree-sha1 = "e2cfc4012a19088254b3950b85c3c1d8882d864d"
uuid = "276daf66-3868-5448-9aa4-cd146d93841b"
version = "2.3.1"

    [deps.SpecialFunctions.extensions]
    SpecialFunctionsChainRulesCoreExt = "ChainRulesCore"

    [deps.SpecialFunctions.weakdeps]
    ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"

[[deps.Statistics]]
deps = ["LinearAlgebra", "SparseArrays"]
uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"
version = "1.9.0"

[[deps.StatsAPI]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "1ff449ad350c9c4cbc756624d6f8a8c3ef56d3ed"
uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0"
version = "1.7.0"

[[deps.StatsBase]]
deps = ["DataAPI", "DataStructures", "LinearAlgebra", "LogExpFunctions", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"]
git-tree-sha1 = "1d77abd07f617c4868c33d4f5b9e1dbb2643c9cf"
uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
version = "0.34.2"

[[deps.StatsFuns]]
deps = ["HypergeometricFunctions", "IrrationalConstants", "LogExpFunctions", "Reexport", "Rmath", "SpecialFunctions"]
git-tree-sha1 = "f625d686d5a88bcd2b15cd81f18f98186fdc0c9a"
uuid = "4c63d2b9-4356-54db-8cca-17b64c39e42c"
version = "1.3.0"

    [deps.StatsFuns.extensions]
    StatsFunsChainRulesCoreExt = "ChainRulesCore"
    StatsFunsInverseFunctionsExt = "InverseFunctions"

    [deps.StatsFuns.weakdeps]
    ChainRulesCore = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4"
    InverseFunctions = "3587e190-3f89-42d0-90ee-14403ec27112"

[[deps.SuiteSparse]]
deps = ["Libdl", "LinearAlgebra", "Serialization", "SparseArrays"]
uuid = "4607b0f0-06f3-5cda-b6b1-a6196a1729e9"

[[deps.SuiteSparse_jll]]
deps = ["Artifacts", "Libdl", "Pkg", "libblastrampoline_jll"]
uuid = "bea87d4a-7f5b-5778-9afe-8cc45184846c"
version = "5.10.1+6"

[[deps.TOML]]
deps = ["Dates"]
uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76"
version = "1.0.3"

[[deps.Tar]]
deps = ["ArgTools", "SHA"]
uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e"
version = "1.10.0"

[[deps.Test]]
deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40"

[[deps.UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

[[deps.Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.13+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.8.0+0"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.48.0+0"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+0"
"""

# ╔═╡ Cell order:
# ╟─5d67bf6a-6a9d-11ee-3858-5be18c658ac6
# ╟─2b227cb3-860a-464f-bba5-220fec035d12
# ╠═dff33215-c1f4-47db-bfd1-24de63c9f67d
# ╠═81bb0432-587e-4921-8da8-a3fcfbb8325d
# ╠═6f2b0ba5-e30f-4e63-a42b-b9614b41d897
# ╟─929a5549-0e8a-4df3-b378-b74cef110bd0
# ╠═31f3abea-6d66-488c-a9b6-6b3fdb733447
# ╟─34e69d0c-980f-4d45-bbbf-6241a9061be5
# ╠═20d3ddda-d10e-42d2-b05b-b8af1b9cad5e
# ╠═36957cfc-a427-43b1-9999-7fedc584e789
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002