{
"cells": [
{
"cell_type": "code",
"execution_count": 23,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
""
],
"text/plain": [
""
]
},
"execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# nbi:hide_in\n",
"\n",
"from IPython.display import display, HTML\n",
"\n",
"CSS = \"\"\"\n",
".output {\n",
" align-items: center;\n",
"}\n",
"\"\"\"\n",
"HTML(''.format(CSS))\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "slide"
}
},
"source": [
"# 1: Introduction, probability, combinatorics\n",
"\n",
" #### [Back to main page](https://petrosyan.page/fall2020math3215)\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"slideshow": {
"slide_type": "skip"
}
},
"source": [
"## Outcomes and events"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Suppose we are observing a phenomenon with a certain outcome (an experiment, the fluctuations of the stock market, etc). Let $S$ denote the space of all possible outcomes of this phenomenon. One such example is the toss of a coin. Each toss results in an outcome of either H (head) or T (tail) so in this case, the outcome space will be $S=\\{H, T\\}$. Or if we are pulling a card from a deck then the space of all outcomes is the set of all individual cards. \n",
"\n",
"Often times, we are not interested in a specific outcome but rather if the outcome has certain qualities. In the case of the pulled card, we may be interested if it is spades rather than being a specific card from spades. In other words, we are often interested in sets (collections) of outcomes. Such sets we conventionally call events in probability theory. \n",
"\n",
"* If we want the outcome to have quality $A$ and quality $B$ simultaneously, that will correspond to the event $A\\cap B$.\n",
"* If we want the outcome to have quality $A$ and not have the quality $B$, it corresponds to the event $A\\setminus B$.\n",
"* If we want the outcome to either have quality $A$ or $B$ then it corresponds to the event $A\\cup B$.\n",
"* If we want the outcome to not have quality $A$ then it corresponds to the event $A^\\prime = S\\setminus A$.\n",
"\n",
"\n",
"**Definition:** \n",
"\n",
"\n",
"
\n",
"\n",
"1. A set of events $A_1,\\dots, A_n$ are called mitually exclusive if $A_i\\cap A_j=\\emptyset$ for every $i\\neq j$. \n",
"2. A set of events $A_1,\\dots, A_n$ are called exhaustive if\n",
" \n",
" $$\\displaystyle\\bigcup_{i=1}^nA_i=S.$$\n",
"\n",
"
\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### De Morgan's rules\n",
"\n",
"For any two events $A$ and $B$\n",
"\n",
"1. $(A\\cup B)^\\prime = A^\\prime \\cap B^\\prime $,\n",
"2. $(A\\cap B)^\\prime = A^\\prime \\cup B^\\prime $.\n",
"\n",
"(to convince yourself, see the Venn diagram below)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"![demorgan](images/demor.png)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Probability"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"There are many situations where a repetitive phenomenon takes place each time with a different outcome. In this case, we say that the outcome is random. The probability of an outcome is a number between 0 and 1 that shows the relative frequency at which the random phenomenon results in that outcome. \n",
"\n",
"Similarly, the probability of an event $A$ will represent the proportion of outcomes that terminated in $A$.\n",
"\n",
"For example, consider the coin toss experiment. If we were to repeat the toss many times there is no inherent reason to expect that the number of heads should be dominating over the number of tails or vice versa. I have experimented and recorded the outcome of hundred such tosses shown below (used a quarter).\n",
"\n",
"The relative frequency of heads after $i$ tosses is \n",
"$$f(H) = \\frac{\\text{# of heads after i tosses}}{i}.$$\n",
"\n",
"The outcomes were \n",
"\n",
" [T,T,T,H,H,H,H,T,T,T,H,T,T,T,T,T,H,H,H,T,T,H,H,T,H,\n",
" H,T,H,T,H,H,T,T,H,T,T,T,H,H,H,H,T,T,H,T,T,T,T,T,T,\n",
" H,T,T,H,H,H,T,T,T,H,T,H,T,H,H,H,T,T,H,H,H,H,T,H,H,\n",
" T,T,T,T,T,H,T,H,H,T,T,H,H,H,T,H,T,H,H,H,T,T,T,T,T]\n",
" \n",
"The graph below shows the frequency as I increase the number of samples. If I repeated this experiment for long enough the graph would eventually stabilize and become closer to $1/2$."
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
"outputs": [
{
"data": {
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\n",
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