# Foundations of probability theory: basic definitions

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This module covers the basic ideas of Probability Theory. It reviews the laws of boolean algebra, describes how to compute a priori and conditional probabilities, and uses these properties to obtain Bayes' Rule.

## Basic definitions

The basis of probability theory is a set of events - sample space - and a systematic set of numbers - probabilities -assigned to each event. The key aspect of the theory is the system of assigning probabilities. Formally, a sample space is the set  of all possible outcomes ${}_{i}$ of an experiment. An event is a collection of sample points ${}_{i}$ determined by some set-algebraic rules governed by the laws of Boolean algebra.Letting $A$ and $B$ denote events, these laws are $\text{Union:}A\cup B=\{\colon \in A\lor \in B\}$ $\text{Intersection:}A\cap B=\{\colon \in A\land \in B\}$ $\text{Complement:}(A)=\{\colon \notin A\}$ $(A\cup B)=(A)\cap (B)$ The null set $\emptyset$ is the complement of  . Events are said to be mutually exclusive if there is no element common to both events: $A\cap B=\emptyset$ .

Associated with each event ${A}_{i}$ is a probability measure $({A}_{i})$ , sometimes denoted by ${}_{i}$ , that obeys the axioms of probability .

• $({A}_{i})\ge 0$
• $()=1$
• If $A\cap B=\emptyset$ , then $(A\cup B)=(A)+(B)$ .
The consistent set of probabilities $()$ assigned to events are known as the a priori probabilities . From the axioms, probability assignments for Boolean expressions can becomputed. For example, simple Boolean manipulations ( $A\cup B=A\cup (A)B$ ) lead to
$(A\cup B)=(A)+(B)-(A\cap B)$

Suppose $(B)\neq 0$ . Suppose we know that the event $B$ has occurred; what is the probability that event $A$ has also occurred? This calculation is known as the conditional probability of $A$ given $B$ and is denoted by $(B, A)$ . To evaluate conditional probabilities, consider $B$ to be the sample space rather than  . To obtain a probability assignment under these circumstances consistentwith the axioms of probability, we must have

$(B, A)=\frac{(A\cap B)}{(B)}$
The event is said to be statistically independent of $B$ if $(B, A)=(A)$ : the occurrence of the event $B$ does not change the probability that $A$ occurred. When independent, the probability of their intersection $(A\cap B)$ is given by the product of the a priori probabilities $(A)(B)$ . This property is necessary and sufficient for the independence of the two events. As $(B, A)=\frac{(A\cap B)}{(B)}$ and $(A, B)=\frac{(A\cap B)}{(A)}$ , we obtain Bayes' Rule .
$(A, B)=\frac{(B, A)(B)}{(A)}$

what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years? Kala Reply lim x to infinity e^1-e^-1/log(1+x) given eccentricity and a point find the equiation Moses Reply 12, 17, 22.... 25th term Alexandra Reply 12, 17, 22.... 25th term Akash College algebra is really hard? Shirleen Reply Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table. Carole I'm 13 and I understand it great AJ I am 1 year old but I can do it! 1+1=2 proof very hard for me though. Atone hi Adu Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily. Vedant find the 15th term of the geometric sequince whose first is 18 and last term of 387 Jerwin Reply I know this work salma The given of f(x=x-2. then what is the value of this f(3) 5f(x+1) virgelyn Reply hmm well what is the answer Abhi If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10 Augustine how do they get the third part x = (32)5/4 kinnecy Reply make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be AJ how Sheref can someone help me with some logarithmic and exponential equations. Jeffrey Reply sure. what is your question? ninjadapaul 20/(×-6^2) Salomon okay, so you have 6 raised to the power of 2. what is that part of your answer ninjadapaul I don't understand what the A with approx sign and the boxed x mean ninjadapaul it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared Salomon I'm not sure why it wrote it the other way Salomon I got X =-6 Salomon ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6 ninjadapaul oops. ignore that. ninjadapaul so you not have an equal sign anywhere in the original equation? ninjadapaul hmm Abhi is it a question of log Abhi 🤔. Abhi I rally confuse this number And equations too I need exactly help salma But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends salma Commplementary angles Idrissa Reply hello Sherica im all ears I need to learn Sherica right! what he said ⤴⤴⤴ Tamia hii Uday hi salma hi Ayuba Hello opoku hi Ali greetings from Iran Ali salut. from Algeria Bach hi Nharnhar what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks. Kevin Reply a perfect square v²+2v+_ Dearan Reply kkk nice Abdirahman Reply A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place. Kimberly Reply Jeannette has$5 and \$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
What is the expressiin for seven less than four times the number of nickels
How do i figure this problem out.
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Shanjida
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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