COMP 2711H: Lecture 32

Date: 2024-11-13 04:45:44

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Probability Theory

Probability theory
- $n$ experiments (coin flip): will result in
  - $h$ heads
  - $n - h$ tails
- and, as we keep tossing coin's what's the chance of getting a head?
  - $lim_{n \to \infty} \frac{h}{n} = ?$
  - 👨‍🏫 can we use the limit for probability? probably not
    - limit: might not even exist!
Monty Hall problem
- one of the doors: have a prize
  - selection: in uniformly random
- you choose one door. then, the host opens an empty door that you didn't choose
  - then, is it better to change your choice?
- w/ computer simulation: you win w/ $1/3$ without changing
  - and $2/3$ with changing
Sleeping Beauty
- on Monday: Amir tosses a fair coin
  - makes us fall a sleep, w/ medicine that also erases your memory
  - if head: wakes us up on Tuesday
  - if tail: wakes us up on Tuesday
    - then give medicine again, waking us up on Wednesday
- if you compute the probability based on no. of "wake up"-s
  - the chance: seems like $1/3$ , $2/3$ for head and tail
    - due to our non-rigorous definition of probability
  - 👨‍🏫 philosophy people still talk about this
Cancer test
- a novel cancer test: accuracy is as of following
  - if you have cancer: you get + w/ 90 percent
  - if you don't have cancer: you get - w/ 90 percent
- you: take test and gets +, what's the chance that you actually have a cancer?
  - 👨‍🏫 solution: depend on portion of population w/ cancer

Measure

Measure
- what is: measure of every subset?
  - and we can: assign measure - a weight - on each subset
- first: we have extended $R$ : $R \cup {\infty, - \infty}$
  - then positive extended $R$ : $\overset{ˉ}{R} = [0, \infty) \cup {\infty}$
- let $A$ : a set (universe) and $E \subseteq A$
  - measure $m$ : a function $E \to \overset{ˉ}{R}$
- $(A, E, m)$ : a measure space if:
  1. $ϕ \in E$ , and $m (ϕ) = 0$
  2. let $X_{1}, X_{2}, \dots$ be a countable set of pairwise disjoint sets
    - each: $\in E$ , thus measurable
    - $⋃_{i = 0}^{\infty} X_{i} \in E$
    - $m (⋃_{i = 0}^{\infty} X_{i}) = \sum_{i = 0}^{\infty} m (X_{i})$
    - 👨‍🏫 with this only, you can't compute $m (X^{c})$ from $m (C)$
  3. if $X \in E$ , then $X^{c} = A - X \in E$
    - from second axiom: $m (X) + m (X^{c}) = m (S)$
- Lebesgue measure
  - for every segment from $a, b$ : the measure is $b - a$
  - $m (R) = ⋃_{i \in Z} [i, i + 1) = \infty$
    - i.e. length of the whole real number: $\infty$
Set & sample space
- let $S$ : set sample space
  - 👨‍🏫: convention: uses $Ω$ actually
  - sample space: all possibilities
- $E \subseteq P (S)$ events
  - usually: $E = S$
- define function $P r : E \to [0, 1]$
- $(S, E, P r)$ : a probability space if:
  1. $ϕ \in E, S \in E$
  2. $\forall X_{i} \in E, ⟹ ⋃ X_{i} \in E$
  3. $X^{c} = S - X \in E$
  4. $P r (s) = 1$
  5. if $X_{i} \in E$ being disjoint:
    - $P r (⋃ X_{i}) = \sum P r (X_{i})$
- from 4, 5: u=you can derive
  - $P (A) + P (A^{c}) = 1$
Conditional probability
- suppose: we know $S^{'} \subseteq S$
- then: the information might change the probability
  - at least: we cant' say it persists
  - as $P r (S) = 1$ regardless
- definition: let $A, B \in E$
  - define: $P r (A ∣ B) = \frac{P r ( A \cap B )}{P r ( B )}$
  - we are defining: a new prob. function
  - better notation: $P_{B} (A) = \frac{P r ( A \cap B )}{P r ( B )}$

COMP 2711H: Honors Discrete Mathematical Tools for Computer Science