A TON OF NEW TEXTBOOKS

2026-06-04 21:16:45 -07:00
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+   "source": [
+    "import os\n",
+    "import sys\n",
+    "\n",
+    "import matplotlib.pyplot as plt\n",
+    "import numpy as np\n",
+    "import pandas as pd\n",
+    "import seaborn as sns\n",
+    "\n",
+    "sns.set_theme(style=\"whitegrid\", context=\"notebook\")"
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+   "id": "a6732353-51d5-4478-9cf8-5834e57e5a4e",
+   "metadata": {},
+   "source": [
+    "# Chapter 2 Notes"
+   ]
+  },
+  {
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+   "id": "9f0046c2",
+   "metadata": {},
+   "source": [
+    "## Counting (2.0.0 - 2.1.5)\n",
+    "\n",
+    "***Definition.*** Multiplication Principle:  \\\n",
+    "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",
+    "\n",
+    "### Terminology\n",
+    "\n",
+    "- **Sampling**: Sampling from a set means choosing an element from that set. We\n",
+    "often **draw** a sample at random from a given set in which each element of the\n",
+    "set has equal chance of being chosen\n",
+    "\n",
+    "- **With or without replacement**: Usually we draw multiple samples from a set. If\n",
+    "we put each object back after each draw, we call this sampling with\n",
+    "replacement. In this case a single object can be possibly chosen multiple times.\n",
+    "For example, if A = {a1, a2, a3, a4} and we pick 3 elements with replacement, a\n",
+    "possible choice might be (a3, a1, a3). Thus \"with replacement\" means \"repetition\n",
+    "is allowed.\" On the other hand, if repetition is not allowed, we call it sampling\n",
+    "without replacement\n",
+    "\n",
+    "- **Ordered or unordered**: If ordering matters (i.e.: a1, a2, a3 ≠ a2, a3, a1), this is\n",
+    "called ordered sampling. Otherwise, it is called unordered\n",
+    "\n",
+    "### Counting Formulas\n",
+    "\n",
+    "- **ordered sampling with replacement:** $n^k$\n",
+    "\n",
+    "- **ordered sampling without replacement:** $n$ permute $k$ $\\quad$ ie $P^n_k = \\frac{n!}{(n - k)!}$\n",
+    "\n",
+    "- **unordered sampling without replacement:** $n$ choose $k$ $\\quad$ ie $\\binom{n}{k} = \\frac{n!}{k!(n-k)!}$\n",
+    "\n",
+    "- **unordered sampling with replacement:** $\\binom{n + k - 1}{k}$"
+   ]
+  }
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