Lunarnotes - SW11

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    "# Joint probabilities\n",
    "## Conditional probabilities\n",
    "\n",
    "- test, selection with replacement\n",
    "\n",
    "### Rules\n",
    "\n",
    "- $\\mathbb{P}(A|B) \\neq \\mathbb{P}(B|A)$\n",
    "\n",
    "- $\\mathbb{P}(A|B) = \\dfrac{\\mathbb{P}(A \\cap B)}{\\mathbb{P}(B)}$\n",
    "\n",
    "- $\\mathbb{P}(A_1 \\cap A_2|B) = \\mathbb{P}(A_1|B)+\\mathbb{P}(A_2|B)$ if $A_1 \\cap A_2 \\cap B = \\emptyset$\n",
    "\n",
    "- $\\mathbb{P}(\\bar{A}|B) = 1-\\mathbb{P}(A|B)$\n",
    "\n",
    "### Bayes' theorem\n",
    "Let A and B be events with $\\mathbb{P}(A) > 0$ and $\\mathbb{P}(B) > 0$. Then, between $\\mathbb{P}(A|B)$ and $\\mathbb{P}(B|A)$ there is the relationship:\n",
    "$$\\mathbb{P}(A|B) = \\dfrac{\\mathbb{P}(A \\cap B)}{\\mathbb{P}(B)} = \\dfrac{\\mathbb{P}(B|A)\\mathbb{P}(A)}{\\mathbb{P}(B)}$$\n",
    "\n",
    "Therefore:\n",
    "\n",
    "$$\\mathbb{P}(B|A) = \\dfrac{\\mathbb{P}(A \\cap B)}{\\mathbb{P}(A)} = \\dfrac{\\mathbb{P}(A|B)\\mathbb{P}(B)}{\\mathbb{P}(A)}$$\n",
    "\n",
    "Solving for $\\mathbb{P}(B)$:\n",
    "$$\\mathbb{P}(B) = \\mathbb{P}(B \\cap A)+\\mathbb{P}(B \\cap \\bar{A})$$\n",
    "\n",
    "\n",
    "## Unconditional probabilities\n",
    "\n",
    "- rolling dice\n",
    "\n",
    "When $B$ is a prerequisit:\n",
    "\n",
    "$\\mathbb{P}(A|B)= \\mathbb{P}(A)$\n",
    "\n",
    "### Prove if the space is conditional\n",
    "\n",
    "If $\\mathbb{P}(A \\cap B) \\neq \\mathbb{P}(A) \\cdot \\mathbb{P}(B)$, so we are in a unconditional space\n",
    "\n",
    "***\n",
    "# Probability trees\n",
    "\n",
    "![Tree](https://upload.wikimedia.org/wikipedia/commons/9/9c/Probability_tree_diagram.svg)\n",
    "\n",
    "***\n",
    "# Partition\n",
    "\n",
    "## Definition\n",
    "The subset $A_1,...,A_k$ of $\\Omega$ are called $\\textit{partition}$ of $\\Omega$ if the following is hold:\n",
    "1. $A_1 \\cup ... \\cup A_k = \\Omega$\n",
    "2. $\\forall i \\neq j,\\ A_i \\cap A_j = \\emptyset$\n",
    "\n",
    "## Law of total probability\n",
    "$\\forall A_1,...,A_k \\in \\Omega \\land \\forall B \\subseteq \\Omega$:\n",
    "\n",
    "$$\n",
    "\\begin{aligned}\n",
    "\\mathbb{P}(B) &= \\mathbb{P}(B|A_1)\\mathbb{P}(A_1) + \\dots + \\mathbb{P}(B|A_k)\\mathbb{P}(A_k) \\\\\n",
    "&= \\sum_{i} \\mathbb{P}(B|A_i)\\mathbb{P}(A_i)\n",
    "\\end{aligned}\n",
    "$$\n",
    "\n",
    "***\n",
    "# Stochastic indipendence\n",
    "Events $A,\\, B$ are called $\\textit{stochastically indipendent}$ if:\n",
    "$$\\mathbb{P}(A \\cap B) = \\mathbb{P}(A)\\mathbb{P}(B)$$"
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SW11_Notes