From 334b28ed945577b12df6805d560d8f605500e82e Mon Sep 17 00:00:00 2001
From: Mark <mark@betalupi.com>
Date: Thu, 23 Jan 2025 19:53:30 -0800
Subject: [PATCH] Added "Somewhat Random Numbers" (#6)

Reviewed-on: https://git.betalupi.com/Mark/handouts/pulls/6
Co-authored-by: Mark <mark@betalupi.com>
Co-committed-by: Mark <mark@betalupi.com>
---
 src/Warm-Ups/Somewhat Random Numbers/main.typ | 81 +++++++++++++++++++
 .../Somewhat Random Numbers/meta.toml         |  6 ++
 2 files changed, 87 insertions(+)
 create mode 100644 src/Warm-Ups/Somewhat Random Numbers/main.typ
 create mode 100644 src/Warm-Ups/Somewhat Random Numbers/meta.toml

diff --git a/src/Warm-Ups/Somewhat Random Numbers/main.typ b/src/Warm-Ups/Somewhat Random Numbers/main.typ
new file mode 100644
index 0000000..a6c82f6
--- /dev/null
+++ b/src/Warm-Ups/Somewhat Random Numbers/main.typ	
@@ -0,0 +1,81 @@
+#import "@local/handout:0.1.0": *
+
+#show: doc => handout(
+  doc,
+  quarter: link(
+    "https://betalupi.com/handouts",
+    "betalupi.com/handouts",
+  ),
+
+  title: [Somewhat Random Numbers],
+  by: "Mark",
+)
+
+#problem()
+Alice generates 100 random numbers uniformly from $[0,1]$. \
+Bob generates 101 random numbers from $[0, 1]$, but deletes the lowest result.
+
+#v(2mm)
+
+Say we have both of the resulting arrays, but do not know who generated each one. \
+We would like to guess which of the two was generated by Bob. \
+What is the optimal strategy, and what is its probability of guessing correctly?
+
+
+#solution([
+  Looking at the mean seems like a good idea, but there's a better way: \
+  Assign the array with the smaller _minimum_ to Alice.
+
+  #v(3mm)
+
+  To compute the probability, generate 201 numbers. \
+  Assign the first 100 to Alice and the rest to Bob. \
+  Look at the lowest two numbers (of these 201, *before* Bob drops his lowest).
+
+  #v(8mm)
+
+  We'll use the following notation: \
+  `AB` means the lowest was owned by Alice, and the second-lowest, by Bob.
+
+  #v(2mm)
+
+  Probabilities are as follows: \
+  - `AA`: $100\/201 times 99\/200 approx 0.246$
+  - `AB`: $100\/201 times 101\/200 approx 0.251$
+  - `BA`: $101\/201 times 100\/200 approx 0.251$ // spell:disable-line
+  - `BB`: $101\/201 times 100\/200 approx 0.251$
+
+  #v(4mm)
+  Now, Bob drops his lowest number. \
+  We'll cross out the number he drops and box the new lowest number (i.e, the one we observe):
+  - #{
+      (
+        box(`A`, stroke: ored, inset: 1pt)
+          + box(`A`, inset: 1pt)
+          + box([: $approx 0.246$], inset: (top: 1pt, bottom: 1pt))
+      )
+    }
+  - #{
+      (
+        box(`A`, stroke: ored, inset: 1pt)
+          + box(strike(`B`), inset: 1pt)
+          + box([: $approx 0.251$], inset: (top: 1pt, bottom: 1pt))
+      )
+    }
+  - #{
+      (
+        box(strike(`B`), inset: 1pt)
+          + box(`A`, stroke: ored, inset: 1pt)
+          + box([: $approx 0.251$], inset: (top: 1pt, bottom: 1pt))
+      )
+    }
+  - #{
+      (
+        box(strike(`B`), inset: 1pt)
+          + box(`B`, stroke: ored, inset: 1pt)
+          + box([: $approx 0.251$], inset: (top: 1pt, bottom: 1pt))
+      )
+    }
+  #v(8mm)
+  Alice has the smallest number in 3 of 4 cases, which have a total probability of $approx 0.749$.
+])
diff --git a/src/Warm-Ups/Somewhat Random Numbers/meta.toml b/src/Warm-Ups/Somewhat Random Numbers/meta.toml
new file mode 100644
index 0000000..974af61
--- /dev/null
+++ b/src/Warm-Ups/Somewhat Random Numbers/meta.toml	
@@ -0,0 +1,6 @@
+[metadata]
+title = "Somewhat Random Numbers"
+
+[publish]
+handout = true
+solutions = true