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sim_simple.jl
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sim_simple.jl
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### A Pluto.jl notebook ###
# v0.18.0
using Markdown
using InteractiveUtils
# ╔═╡ e1ee796a-2e11-11ec-0fdc-a3b6c2ae2251
using Random, StatsBase, Statistics, CairoMakie, ColorSchemes, Colors
# ╔═╡ 37af6f76-aa8c-4c7f-8705-e55101f22186
begin
cool_theme = Theme(
palette = (color=[c for c in ColorSchemes.seaborn_muted6], marker=[:circle, :utriangle, :cross, :rect, :diamond, :dtriangle, :pentagon, :xcross]),
textcolor = :gray40,
linewidth=4,
fontsize=20,
# font="open-sans",
resolution = (550, 400),
Axis = (
backgroundcolor = RGB(1.0, 1.0, 1.0),
xgridcolor = (:black, 0.15),
ygridcolor = (:black, 0.15),
xminorgridcolor = (:gray, 0.15),
yminorgridcolor = (:gray, 0.15),
leftspinevisible = false,
rightspinevisible = false,
ygridstyle=:dash,
xgridstyle=:dash,
bottomspinevisible = false,
topspinevisible = false,
xminorticksvisible = false,
yminorticksvisible = false,
xticksvisible = false,
yticksvisible = false,
xlabelpadding = 3,
ylabelpadding = 3
),
Legend = (
framevisible = true,
titlehalign=:left,
titlesize=16,
labelsize=16,
framecolor=(:black, 0.5)
# padding = (1, 0, 0, 0),
),
Axis3 = (
xgridcolor = (:black, 0.07),
ygridcolor = (:black, 0.07),
zgridcolor = (:black, 0.07),
xspinesvisible = false,
yspinesvisible = false,
zspinesvisible = false,
xticksvisible = false,
yticksvisible = false,
zticksvisible = false,
),
Colorbar = (
ticksvisible = false,
spinewidth = 0,
ticklabelpad = 5,
)
)
set_theme!(cool_theme)
end
# ╔═╡ 91e5f121-1224-42a9-978b-5b087d9e8d78
include_sims_in_viz = false
# ╔═╡ bf71ffc2-e3b3-494f-beb1-12fdb012cb3c
begin
struct Bridge
L::Int
safe_panels::Array{Int, 1}
observed::Array{Bool, 1}
end
function Bridge(L::Int)
safe_panels = [sample(1:2) for col = 1:L]
observed = [false for col in 1:L]
return Bridge(L, safe_panels, observed)
end
end
# ╔═╡ dc80351f-7048-4a66-bf57-f6beec8887b0
function hop!(bridge::Bridge, c::Int)
if bridge.observed[c]
return true
else
panel_to_hop_on = sample(1:2)
bridge.observed[c] = true
if bridge.safe_panels[c] == panel_to_hop_on
return true
else
return false
end
end
end
# ╔═╡ 263282de-36a7-4b67-9041-0896f94e1220
function traverse!(bridge::Bridge)
for c = 1:bridge.L
survived = hop!(bridge, c)
if ! survived
return false
end
end
return true
end
# ╔═╡ 172949e2-4c23-4f9a-afa2-64df4eff8e6e
# return number of players that were eliminated
function simulate(L::Int, N::Int)
bridge = Bridge(L)
for p = 1:N
survived = traverse!(bridge)
if survived
return p - 1
end
end
return N # if got this far, all eliminated
end
# ╔═╡ 89e25740-5402-4dc6-868e-9e49dd08bb6c
begin
N = 16
L = 32
nb_sims = 100000
sim_nb_eliminated = [simulate(L, N) for i = 1:nb_sims]
sim_probs_survival = zeros(N)
for i = 1:N
sim_probs_survival[i] = sum(sim_nb_eliminated .< i) / nb_sims
end
sim_probs_survival
sim_probs_survival_err = 2 * sqrt.(sim_probs_survival .* (1 .- sim_probs_survival)) / nb_sims
end
# ╔═╡ cdf0ca6e-46ba-411c-b128-2baea0f6aa07
begin
sim_nb_survived = N .- sim_nb_eliminated
Eθ_sim = mean(sim_nb_survived)
end
# ╔═╡ fbb1fd4e-90b5-4bf3-9972-0afc5a52c600
# prob exactly n eliminated
function P_Eₙ(n::Int, L::Int, N::Int)
if n > L
return 0.0
else
if n == N && N < L
return sum([binomial(c-1, N-1) * 0.5 ^ c for c = N:L])
else
return binomial(L, n) * 0.5 ^ L
end
end
end
# ╔═╡ aa91c2ba-0368-4fe5-9d2b-46422855325b
binomial(L, 0)
# ╔═╡ e6b19b92-4e1c-4628-bdba-c2319c96e2f4
sum([P_Eₙ(n, L, N) for n =0:N])
# ╔═╡ edcdbd35-0dbb-447d-b9ed-9499e4dfe895
function P_Sᵢ(i, L, N)
s = 0.0
for j = 0:(i-1)
s += P_Eₙ(j, L, N)
end
return s
end
# ╔═╡ e922b477-b2ed-4fe8-b243-568e832ac933
probs_survival = [P_Sᵢ(i, L, N) for i = 1:N]
# ╔═╡ 57f5ff3f-312e-4f27-a25e-19a71eb893e1
begin
fig = Figure()
# first plot - frequency distribution
ax1 = Axis(
fig[1, 1],
title="N = $N, L = $L",
xlabel="player, i",
ylabel="probability of survival, P(Sᵢ)",
xticks=1:N,
yticks=[0.25 * i for i = 0:4]
)
ylims!(-.01, 1.01)
xlims!(0.5, N+0.5)
barplot!(1:N, probs_survival,
color=ColorSchemes.Derain[2], label="theory")
if include_sims_in_viz
scatter!(1:N, sim_probs_survival,
color="black", markersize=12, label="simulation")
errorbars!(1:N, sim_probs_survival, sim_probs_survival_err, sim_probs_survival_err, color=:red)
axislegend(position=:lt)
end
current_figure()
end
# ╔═╡ 59c7692a-4bcf-4ced-8f44-07c86414276c
save("sim_vs_theory_N$(N)_L$(L).pdf", fig)
# ╔═╡ d0fe1298-8550-4efc-affb-11e7b104cc87
function Eθ(L::Int, N::Int)
θ = 0.0
for n = 0:N
θ += (N - n) * P_Eₙ(n, L, N)
end
return θ
end
# ╔═╡ c4ec3b9c-7157-4b80-9c71-10e9f5307daa
Eθ(L, N)
# ╔═╡ 84dd0f2c-8b35-4512-a996-d29ed44fab15
begin
Ls = 0:50
fig2 = Figure()
# first plot - frequency distribution
ax2 = Axis(
fig2[1, 1],
title="N = $N",
xlabel="bridge length, L",
ylabel="expected # of survivors, 𝔼[θ]",
xticks=[0,10,20,30,40,50],
yticks=[4*i for i = 0:4]
)
xlims!(-0.4, maximum(Ls)+0.4)
ylims!(-0.2, nothing)
hlines!(ax2, N, color="gray", label="all survive", linestyle=:dash)
barplot!(Ls, [Eθ(L_i, N) for L_i in Ls], linewidth=6, color=ColorSchemes.Derain[5])
# hlines!(ax2, 0, color="black", label="none survive",linestyle=:dash)
axislegend(position=:rc)
fig2
end
# ╔═╡ 12be8d50-cad6-4dd6-8a97-ce13b0ed3f24
save("expected_theta.pdf", fig2)
# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
CairoMakie = "13f3f980-e62b-5c42-98c6-ff1f3baf88f0"
ColorSchemes = "35d6a980-a343-548e-a6ea-1d62b119f2f4"
Colors = "5ae59095-9a9b-59fe-a467-6f913c188581"
Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"
StatsBase = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
[compat]
CairoMakie = "~0.7.2"
ColorSchemes = "~3.17.1"
Colors = "~0.12.8"
StatsBase = "~0.33.14"
"""
# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
# This file is machine-generated - editing it directly is not advised
[[AbstractFFTs]]
deps = ["ChainRulesCore", "LinearAlgebra"]
git-tree-sha1 = "6f1d9bc1c08f9f4a8fa92e3ea3cb50153a1b40d4"
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version = "1.1.0"
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[[Adapt]]
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git-tree-sha1 = "af92965fb30777147966f58acb05da51c5616b5f"
uuid = "79e6a3ab-5dfb-504d-930d-738a2a938a0e"
version = "3.3.3"
[[Animations]]
deps = ["Colors"]
git-tree-sha1 = "e81c509d2c8e49592413bfb0bb3b08150056c79d"
uuid = "27a7e980-b3e6-11e9-2bcd-0b925532e340"
version = "0.4.1"
[[ArgTools]]
uuid = "0dad84c5-d112-42e6-8d28-ef12dabb789f"
version = "1.1.1"
[[ArrayInterface]]
deps = ["Compat", "IfElse", "LinearAlgebra", "Requires", "SparseArrays", "Static"]
git-tree-sha1 = "745233d77146ad221629590b6d82fe7f1ddb478f"
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[[Artifacts]]
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[[Automa]]
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git-tree-sha1 = "d50976f217489ce799e366d9561d56a98a30d7fe"
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version = "0.8.2"
[[AxisAlgorithms]]
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git-tree-sha1 = "66771c8d21c8ff5e3a93379480a2307ac36863f7"
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[[Base64]]
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git-tree-sha1 = "19a35467a82e236ff51bc17a3a44b69ef35185a2"
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[[CEnum]]
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[[Cairo]]
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git-tree-sha1 = "d0b3f8b4ad16cb0a2988c6788646a5e6a17b6b1b"
uuid = "159f3aea-2a34-519c-b102-8c37f9878175"
version = "1.0.5"
[[CairoMakie]]
deps = ["Base64", "Cairo", "Colors", "FFTW", "FileIO", "FreeType", "GeometryBasics", "LinearAlgebra", "Makie", "SHA", "StaticArrays"]
git-tree-sha1 = "90fe6622efbb627e7c962e9bd6f5c4228680b7ca"
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[[Cairo_jll]]
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git-tree-sha1 = "4b859a208b2397a7a623a03449e4636bdb17bcf2"
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[[ChainRulesCore]]
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git-tree-sha1 = "f9982ef575e19b0e5c7a98c6e75ee496c0f73a93"
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git-tree-sha1 = "12fc73e5e0af68ad3137b886e3f7c1eacfca2640"
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