https://mathwiki.cs.ut.ee/ 2026-05-15T08:18:40+00:00 https://mathwiki.cs.ut.ee/ https://mathwiki.cs.ut.ee/lib/exe/fetch.php?media=wiki:dokuwiki.svg text/html 2014-01-20T00:19:33+00:00 Anonymous (anonymous@undisclosed.example.com) https://mathwiki.cs.ut.ee/doku.php?id=probability:01_event_probability&rev=1390177173&do=diff Event and probability Let us start with a simple experiment by rolling a normal six-faced dice on a completely flat ground. The outcome can be one, two, three, four, five (like in the picture), or six. If we roll the dice, one of these six outcomes always appears, i.e. every result of the dice roll can be found in the list. And each one of these outcomes excludes the others, e. g., if we roll a five, we could not get in the same time two or six or any other of the listed outcomes (we … text/html 2014-03-06T11:22:42+00:00 Anonymous (anonymous@undisclosed.example.com) https://mathwiki.cs.ut.ee/doku.php?id=probability:02_multiple_event_probability&rev=1394104962&do=diff Probability of two events 1. Mutually exclusive events If the probability distribution of an experiment/process is given, finding the probability of any event is really simple due to the law of mutually exclusive events. The law of mutually exclusive events.$A$$B$$A$$B$$\text{Pr}[A\dot{\cup} B]$$\text{Pr}[A]$$\text{Pr}[B]$$$\text{Pr}[A\dot{\cup} B]=\text{Pr}[A]+\text{Pr}[B].$$$A$$B$$A\cap B=\emptyset$$A$$B$$A\dot{\cup} B$$A$$B$$A$$B$$A{\cup} B$$A$$B$$A$$B$$A{\cap} B$$A$$B$$C:=\text{"we rol… text/html 2014-01-20T13:57:09+00:00 Anonymous (anonymous@undisclosed.example.com) https://mathwiki.cs.ut.ee/doku.php?id=probability:03_conditional_probability&rev=1390226229&do=diff Conditional probability So far, we have seen how probabilities of events are modeled if these events happen without any conditions, e.g. probability of rolling a six, or probability that tomorrow it rains. However, often one has to ask “what if$\text{Pr}[A|B]$$A$$B$$\text{Pr}[x+y=12 | x=6]=\frac{1}{6}$$x,y$$x=6$$y=6$$x=y=6$$x+y=12$$x$$\text{Pr}[x+y=12]=\text{Pr}[x=6]\cdot\text{Pr}[y=6]=\frac{1}{36}$$t>10$$w=\text{sun}$$\text{Pr}[t>10|w=\text{sun}]$$\text{Pr}[t>10|w=\text{sun}] > \text{Pr}[t>… text/html 2014-01-20T13:57:44+00:00 Anonymous (anonymous@undisclosed.example.com) https://mathwiki.cs.ut.ee/doku.php?id=probability:04_total_probability&rev=1390226264&do=diff Total probability Bayes' law is tightly related with the law of total probability. For example, let us view the exercise form previous lesson: We have two urns, $\text{I}$ and $\text{II}$. Urn $\text{I}$ contains $2$ black balls and $3$ white balls. Urn $\text{II}$ contains $1$ black ball and $1$ white ball. An urn is drawn at random and a ball is chosen at random from it. What is the probability of choosing urn $\text{I}$$\text{Pr}[B|\text{I}]=\frac{2}{5}$$B:=\text{"a black ball is drawn"}$… text/html 2014-01-20T14:05:35+00:00 Anonymous (anonymous@undisclosed.example.com) https://mathwiki.cs.ut.ee/doku.php?id=probability:05_expected_value&rev=1390226735&do=diff Expected value Let the probability distribution of a nonstandard two-sided coin toss be as follows $A$ $\text{"heads on top"}$ $\text{"tails on top"}$ $\text{Pr}[A]$ $\frac{2}{3}$ $\frac{1}{3}$ Let us have a game. If the coin shows heads, the bank will give you $2$ € and if the coin shows tails, you have to give to the bank $3$$\text{"heads on top"}\longmapsto 2$$\text{"tails on top"}\longmapsto -3$$X:\Omega\longrightarrow\mathbb{R}$$\omega\in\Omega$$X(\omega… text/html 2014-01-31T16:59:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://mathwiki.cs.ut.ee/doku.php?id=probability:06_inequalities&rev=1391187596&do=diff Inequalities Sometimes it is impossible or not reasonable to calculate the precise value of a probability. In such a case the following inequalities will come in handy. 1. Markov's inequality Markov's inequality gives an upper bound for the probability that a non-negative random variable is greater then or equal to some positive constant.$X$$$\text{Pr}[X\geq \alpha]\leq \frac{E(X)}{\alpha}$$$\alpha$$X$$x$$x_1$$x_2$$x_3$$x_4$$x_5$$x_6$$x_7$$\text{Pr}[X=x]$$p_1$$p_2$$p_3$$p_4$$p_5$$p_6$$p_… text/html 2014-01-29T15:30:56+00:00 Anonymous (anonymous@undisclosed.example.com) https://mathwiki.cs.ut.ee/doku.php?id=probability:07_game_notation&rev=1391009456&do=diff Game notation A group of people starts to play the board game "Snakes an ladders". A question arises: how many dice rolls end the game. Clearly the number of rolls can be viewed as random variable. So how to describe this random variable? It depends on number of players, results of dice rolls, location of ladders and snakes. To calculated the distributions of this random variable seams really complicated.$Y$$\ x \pmod{2}\ $$x$$$\begin{array}{l} G_1\\ \left[\begin{array}{l} x\underset{u}\leftar…