moving stuff around
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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 2,
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"id": "c58309b2",
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"metadata": {},
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"outputs": [],
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"source": [
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"import os\n",
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"import sys\n",
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"\n",
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"import matplotlib.pyplot as plt\n",
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"import numpy as np\n",
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"import pandas as pd\n",
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"import seaborn as sns\n",
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"\n",
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"sns.set_theme(style=\"whitegrid\", context=\"notebook\")"
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]
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},
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{
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"cell_type": "markdown",
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"id": "a6732353-51d5-4478-9cf8-5834e57e5a4e",
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"metadata": {},
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"source": [
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"# Chapter 2 Notes"
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]
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},
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{
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"cell_type": "markdown",
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"id": "9f0046c2",
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"metadata": {},
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"source": [
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"## Counting (2.0.0 - 2.1.5)\n",
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"\n",
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"***Definition.*** Multiplication Principle: \\\n",
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"Suppose that we perform $r$ experiments such that the $k\\text{th}$ experiment has $n_k$ possible outcomes, for $k=1,2,\\dots,r$. Then there are a total of $n_1 \\times n_2 \\times n_3 \\times \\dots \\times n_r$ possible outcomes for the sequence of $r$ experiments\n",
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"\n",
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"### Terminology\n",
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"\n",
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"- **Sampling**: Sampling from a set means choosing an element from that set. We\n",
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"often **draw** a sample at random from a given set in which each element of the\n",
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"set has equal chance of being chosen\n",
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"\n",
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"- **With or without replacement**: Usually we draw multiple samples from a set. If\n",
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"we put each object back after each draw, we call this sampling with\n",
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"replacement. In this case a single object can be possibly chosen multiple times.\n",
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"For example, if A = {a1, a2, a3, a4} and we pick 3 elements with replacement, a\n",
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"possible choice might be (a3, a1, a3). Thus \"with replacement\" means \"repetition\n",
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"is allowed.\" On the other hand, if repetition is not allowed, we call it sampling\n",
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"without replacement\n",
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"\n",
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"- **Ordered or unordered**: If ordering matters (i.e.: a1, a2, a3 ≠ a2, a3, a1), this is\n",
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"called ordered sampling. Otherwise, it is called unordered\n",
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"\n",
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"### Counting Formulas\n",
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"\n",
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"- **ordered sampling with replacement:** $n^k$\n",
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"\n",
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"- **ordered sampling without replacement:** $n$ permute $k$ $\\quad$ ie $P^n_k = \\frac{n!}{(n - k)!}$\n",
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"\n",
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"- **unordered sampling without replacement:** $n$ choose $k$ $\\quad$ ie $\\binom{n}{k} = \\frac{n!}{k!(n-k)!}$\n",
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"\n",
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"- **unordered sampling with replacement:** $\\binom{n + k - 1}{k}$"
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]
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},
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{
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"cell_type": "markdown",
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"id": "2e93e0fe",
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"metadata": {},
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"source": [
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"## Problem Solving Principles"
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]
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},
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{
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"cell_type": "markdown",
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"id": "87279e59",
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"metadata": {},
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"source": [
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"When solving a combinatorics problem, consider:\n",
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"1. Does order matter?\n",
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" - Yes → Permutations\n",
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" - No → Combinations\n",
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"\n",
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" - \"Are HHHTT and THHHT the same outcome to me?\"\n",
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"\n",
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"2. Are we sampling with of without replacement?\n",
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" - Without replacement → Hypergeometric (phone problem)\n",
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" - With replacement → Binomial (coin flips)\n",
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" \n",
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" \"Can the same item be chosen twice?\"\n",
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"3. Are the \"groups\" labeled or unlabeled?\n",
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" - Labeled/distinguishable → Just multiply combinations\n",
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" - Unlabeled/interchangeable → Divide by k!\n",
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"\n",
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" \"Does it matter which group is called group 1?\"\n",
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"4. Are the items distinguishable?\n",
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" - Distinguishable → Each item is unique, classical probability applies\n",
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" - Indistinguishable → Outcomes are not equally likely, be careful\n",
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"\n",
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" \"Could I label these items 1 to n?\"\n",
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"5. Is complement of inclusion-exclusion easier?\n",
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" - Complement → When \"at least\" or \"at most\" language appears\n",
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" - Inclusion-Exclusion → When events overlap\n",
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"\n",
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" \"Is the opposite event simpler to count?\"\n",
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"6. Am I counting each outcome exactly once? \n",
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" - If yes, done. Otherwhise we are overcounting or undercounting"
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]
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},
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{
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"cell_type": "markdown",
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"id": "9c5783dc",
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"metadata": {},
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"source": []
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": "roadmap (3.14.5)",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.14.5"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 5
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}
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