OpenLLM-France/wikimedia huggingface dataset:
# Pokémon/Pokédex/Skuntank
## Description

A skunk-like pokémon.

## Base Statistics

A Poison and Dark type Pokémon.

## Type Matching

Moves that deal 200% of their damage to Skuntank: ground, psychic, fighting Moves that deal 50% of their damage to Skuntank: grass, poison, ghost, dark Moves that deal 0% of their damage to Skuntank: psychic
# Pokémon/Pokédex/Sinistcha
## Description

A matcha tea inspired pokémon.

## Base Statistics

A Grass and Ghost type Pokémon.

## Type Matching

Moves that deal 200% of their damage to Sinistcha: fire, ice, flying, ghost, dark Moves that deal 50% of their damage to Sinistcha: water, grass, electric, ground Moves that deal 0% of their damage to Sinistcha: normal, fighting

## How to Evolve

Evolves from Poltchageist.

## Notes

Introduced in the Teal Mask DLC for Pokémon Scarlet and Violet.
# Pokémon/Pokédex/Skitty
## Basic Statistics

* National Index: 300
* Johto Index: 300
* Hoenn Index: 61
* Stage: Basic
* Height: 0.6m/2'0"
* Weight: 11kg/24.0lb
* Gender Distribution: 75% Female
* Ability: Cute Charm
* Type: Normal
* Species: Kitten
* Egg Group: Ground/Fairy
* Hatch Steps: 3840
* Catch Rate: 255
* Base Experience: 65
* Max Experience: 800000
| | HP | Attack | Defense | Speed | Special Attack | Special Defense |
| Base Stats (RB/Y) | - | - | - | - | - | - |
| Base Stats (GS/C) | - | - | - | - | - | - |
| Base Stats (RS/FL/E) | 50 | 45 | 45 | 50 | 35 | 35 |
| Base Stats (DP/PT/HGSS) | – | – | – | – | – | – |
| Base Stats (BW/B2W2) | – | – | – | – | – | – |
| Base Stats (XY/ORAS) | – | – | – | – | – | – |
| Base Stats (SM/USUM) | – | – | – | – | – | – |
| Base Stats (Sword/Shield/BDSP) | – | – | – | – | – | – |
| Max Stats (RB/Y) | - | - | - | - | - | - |
| Max Stats (GS/C) | - | - | - | - | - | - |
| Max Stats (RS/FL/E) | 304 | 207 | 207 | 218 | 185 | 185 |
| Max Stats (DP/PT/HGSS) | – | – | – | – | – | – |
| Max Stats (B/W/B2W2) | – | – | – | – | – | – |
| Max Stats (X/Y/ORAS) | – | – | – | – | – | – |
| Max Stats (SM/USUM) | – | – | – | – | – | – |
| Max Stats (Sword/Shield/BDSP) | – | – | – | – | – | – |
| Effort Points (Sword/Shield/BDSP) | – | – | – | – | – | – |

## Type Matching

Move types which are good on Skitty: Fighting Move types which don't effect Skitty: Ghost
All other move types are normal effectiveness on Skitty.

## How to Evolve

To get a Delcatty from Skitty, use the Moon Stone on Skitty.

## Moveset

| Level | Ruby / Sapphire | Fire Red / Leaf Green |
| Start | Growl Tackle | Growl Tackle |
| 3 | Tail Whip | Tail Whip |
| 7 | Attract | Attract |
| 13 | Sing | Sing |
| 15 | Doubleslap | Doubleslap |
| 19 | Assist | Assist |
| 25 | Charm | Charm |
| 27 | Faint Attack | Faint Attack |
| 31 | Covet | Covet |
| 37 | Heal Bell | Heal Bell |
| 39 | Double-Edge | Double-Edge |

### TM/HM

* RS/FL: Attract, Blizzard, Calm Mind, Dig, Double Team, Facade, Flash, Frustration, Hidden Power, Ice Beam, Iron Tail, Protect, Rain Dance, Rest, Return, Safeguard, Secret Power, Shadow Ball, Shock Wave, Solarbeam, Sunny Day, Thunder, Thunderbolt, Toxic, Water Pulse

### Breeding

* RS/FL: Baton Pass, Fake Tears, Helping Hand, Psych Up, Substitute, Tickle, Uproar, Wish

### Other Sources

* Pokémon Box: Pay Day

## Trading Cards

* EX Ruby & Sapphire #44 - Skitty
* EX Ruby & Sapphire #70 - Skitty
* EX Ruby & Sapphire #71 - Skitty
* EX Sandstorm #79 - Skitty
| Basic: Skitty | Stage 1: Delcatty | (No Stage 2) |
# Pokémon/Pokédex/Skiddo
## Description

A goatlike Pokémon.

## Base Statistics

A Grass type Pokémon.

## Type Matching

Moves that deal 200% of their damage to Skiddo: fire, ice, poison, flying, bug Moves that deal 50% of their damage to Skiddo: water, grass, electric, ground
# Pokémon/Pokédex/Skiploom

National Pokédex:

## Basic Statistics

* National Index: 188
* Johto Index: 68
* Hoenn Index: 333
* Stage: Stage 1
* Height: 0.6m/2'0"
* Weight: 1kg/2.0lb
* Gender Distribution: 50% Male
* Ability: Chlorophyll
* Type: Grass/Flying
* Species: Cottonweed
* Egg Group: Fairy/Plant
* Hatch Steps: 5120
* Catch Rate: 120
* Base Experience: 136
* Max Experience: 1059860
* Hold Item in Advance: none
| | HP | Attack | Defense | Speed | Special Attack | Special Defense |
| Base Stats (RB/Y) | - | - | - | - | - | - |
| Base Stats (GS/C) | 55 | 45 | 50 | 80 | 45 | 65 |
| Base Stats (RS/FL/E) | 55 | 45 | 50 | 80 | 45 | 65 |
| Base Stats (DP/PT/HGSS) | – | – | – | – | – | – |
| Base Stats (BW/B2W2) | – | – | – | – | – | – |
| Base Stats (XY/ORAS) | – | – | – | – | – | – |
| Base Stats (SM/USUM) | – | – | – | – | – | – |
| Base Stats (Sword/Shield/BDSP) | – | – | – | – | – | – |
| Max Stats (RB/Y) | - | - | - | - | - | - |
| Max Stats (GS/C) | 313 | 188 | 198 | 258 | 188 | 228 |
| Max Stats (RS/FL/E) | 314 | 207 | 218 | 284 | 207 | 251 |
| Max Stats (DP/PT/HGSS) | – | – | – | – | – | – |
| Max Stats (B/W/B2W2) | – | – | – | – | – | – |
| Max Stats (X/Y/ORAS) | – | – | – | – | – | – |
| Max Stats (SM/USUM) | – | – | – | – | – | – |
| Max Stats (Sword/Shield/BDSP) | – | – | – | – | – | – |
| Effort Points (Sword/Shield/BDSP) | – | – | – | – | – | – |

## Type Matching

Move types which are very good on Skiploom: Ice Move types which are good on Skiploom: Fire, Flying, Poison, Rock Move types which are bad on Skiploom: Fighting, Water Move types which are very bad on Skiploom: Grass Move types which don't effect Skiploom: Ground
All other move types are normal effectiveness on Skiploom.

## Moveset

| Level | Gold / Silver | Crystal | Ruby / Sapphire Emerald XD / Colosseum | Fire Red / Leaf Green |
| Start | Splash Synthesis Tail Whip Tackle | Splash Synthesis Tail Whip Tackle | Splash Synthesis Tail Whip Tackle | Splash Synthesis Tail Whip Tackle |
| 5 | Tail Whip | Synthesis Tail Whip | Synthesis Tail Whip | Synthesis Tail Whip |
| 10 | Tackle | Tackle | Tackle | Tackle |
| 13 | Poisonpowder | Poisonpowder | Poisonpowder | Poisonpowder |
| 15 | Stun Spore | Stun Spore | Stun Spore | Stun Spore |
| 17 | Sleep Powder | Sleep Powder | Sleep Powder | Sleep Powder |
| 22 | Leech Seed | Leech Seed | Leech Seed | Leech Seed |
| 29 | Cotton Spore | Cotton Spore | Cotton Spore | Cotton Spore |
| 36 | Mega Drain | Mega Drain | Mega Drain | Mega Drain |

### TM/HM

* GS/C/RS/FL: Attract, Double Team, Flash, Frustration, Giga Drain, Hidden Power, Protect, Rest, Return, Solarbeam, Sunny Day, Toxic
* GS/C: Curse, Defense Curl, Endure, Headbutt, Sleep Talk, Snore, Swagger, Sweet Scent
* RS/FL: Aerial Ace, Bullet Seed, Facade, Secret Power

### Evolution

* GS/C/RS/FL: Amnesia, Confusion, Double-Edge, Encore, Reflect
* GS/C: Growl, Pay Day
* RS/FL: Psych Up

## Trading Cards

* Neo Genesis #49 - Skiploom
* Neo Revelation #52 - Skiploom
* Aquapolis #60 - Skiploom
* EX Team Rocket Returns $49 - Skiploom
| Basic: Hoppip | Stage 1: Skiploom | Stage 2: Jumpluff |
# Pokémon/Pokédex/Sinistea
## Description

A small teacup Pokémon.

## Base Statistics

A Ghost type Pokémon.

## Type Matching

Moves that deal 200% of their damage to Sinistea: ghost, dark Moves that deal 50% of their damage to Sinistea: poison, bug Moves that deal 0% of their damage to Sinistea: normal, fighting
# Pinyin/Influx

See also
# Pinyin/Injured

See also
# Pinyin/Injure with knife

See also
# Pinyin/Information

Information is the knowledge or facts about someone or something.
xihnxi shih guanyur moourern huoh moouwuh d zhishir huoh shihshir.
# Pinyin/Injure

See also
# Pinyin/Usually

usually: in the way that is usual or normal
tongcharng dy: yiban huoh zhehngcharng d fangshih
# Pinyin/Use

to use: to do something with a machine, a tool, etc. for a particular purpose
yohng: weihle tehdihng muhdih, yohng jiqih, gongjuh, deeng zuoh shihqirng
# Pinyin/Uyghur

See also
# Pinyin/Used as

See also
# Pinyin/Used to

See also
# Pinyin/Utilize

See also
# Pinyin/Vaccine

See also
A vaccine is a biological preparation that improves immunity to a particular disease. A vaccine typically contains an agent that resembles a disease-causing microorganism, and is often made from weakened or killed forms of the microbe, its toxins or one of its surface proteins. The agent stimulates the body's immune system to recognize the agent as foreign, destroy it, and "remember" it, so that the immune system can more easily recognize and destroy any of these microorganisms that it later encounters.
Yìmiáo shì yìtú tōngguò cìjī chǎnshēng kàngtǐ duì yīzhǒng jíbìng xíngchéng miǎnyìlì de rènhé zhìjì. Yìmiáo bāokuò zhūrú mièhuó huò jiǎndú wēishēngwù de hùnxuányè, huòzhě wēishēngwù zhìpǐn huò yǎnshēngwù. Shǐyòng yìmiáo de zuì chángjiàn fāngfǎ shì zhùshè, dàn yě yǒu yīxiē kǒufú huò shǐyòng bíwùjì.
# Pinyin/Useful

See also
# Pinyin/Usual

See also
# Radiation Oncology/Breast/RT toxicity

* Front Page: Radiation Oncology | RTOG Trials
* Breast: Main Page |
* Staging |
* Breast Overview |
* Prevention |
* Benign |
* DCIS |
* LCIS |
* Paget's |
* Phyllodes tumor |
* Early stage |
* Advanced stage |
* Post mastectomy |
* Inflammatory |
* Partial breast irradiation |
* Regional lymphatics |
* Hormonal therapy |
* Chemotherapy |
* RT technique |
* Recurrence |
* Toxicity of RT |
* Randomized |
* NSABP trials
Breast Cancer RT Toxicity
* Skin care
* Lymphedema
* Lung toxicity
* Cardiac toxicity
* Thyroid
# Radiation Oncology/CALGB

* Front Page: Radiation Oncology | RTOG Trials | Randomized Trials
See the CALGB webpage
See also:
# Rebol Programming/Language Features/View

View is a powerful two dimensional compositing graphical engine that was designed to display user interfaces. It takes a face object, which contains a hierarchical description of various graphical elements, and calls the operating system to render them to the user display. Various visual objects, or faces, may have handlers that deal specifically with certain visual events such as mouse overs, clicks, window resizing etc, and these events are linked into Rebol's event handling system with the call to 'do-events.
This means that when a mouse click is detected in a window, and the mouse click is over a button object, View is able to execute the handling code defined by that button that deals with mouse clicks.
Rebol uses a standard face that is used as a blue print for all graphical objects. View needs to use this as the base object as it contains the minimum set of properties needed to render and allow interaction with the operating system. It is defined in system/standard/face and is also referred to by the global variable face.
Let's define a face by templating from the system face as follows:
The default face has an offset of 0x0, has a color of value 200.200.200 and size of 100x100. We've changed it here to add some red with the gradient changing across the face, repositioned and resized it.
How did we know what properties the system face object has? Simply, by typing
To become proficient directly using View requires an intimate knowledge of the various graphic components. For most users wishing to build user interfaces, this degree of detail is not necessary, and fortunately Rebol has a visual interfaces dialect (VID) which greatly simplifies the process.
To build an interface with a text label, and a text entry field, is as simple with VID as
which looks like this:
layout parses the block, and from this, builds a face with a complex set of graphical elements that handle text editing, and mouse overs. The View engine then renders this to the screen.
Although VID is touted as being only an example of a dialect that can be created to build user interfaces, it has, by virtue of being built in, and being simpler than programming directly in View, become the default way to build user interfaces.
There are a number of problems with this approach. Since VID programming greatly simplifies the builing of a user interface, it also hides the complexity behind what happens so that if you wish to do something which is not documented, it does mean examining the source to VID to see how this might be done. Furthermore, VID remains incomplete in its range of graphical interface objects with third party widgets often failing to work well with each other. There is also a lot of redundancy in the code generated by the layout engine since not every facet of an object is needed in every instance, but is still generated. It also lacks an automatic resizing mechanism, and elements have to be repositioned manually.
# Rebol Programming/Third Party/Beer Framework
### About BEER

BEER stands for Block Extensible Exchange in REBOL. It is a framework to implement application protocols and solves the encapsulation, extensibility, and flow control problems. BEER is essentially a REBOL implementation of the BEEP Protocol plus additional features.
BEER offers simple framing, robust flow control, asynchrony and extensibility through profiles which permit direct implementation of the semantics of the problem domain.
It was inspired by IETF RFC 3080 and 3081, and it is fully asynchronous allowing for non-blocking user-interfaces.
Each API is called a Profile.
Principles and API of BEER foundation are documented here.

#### Async Operation

BEER can give you fully asynchronous operation, which means you can do GUI operations and networking operations simultaneously.
# Rebol Programming/Third party tutorials

* Tutorial for Beginners, by Nick Antonaccio
* Rebol Programming Tutorial, by Nick Antonaccio
* The Rebol scripting language, an introductory article by Carl Sassenrath
* A Beginner's Guide to Rebol Visual Interfaces, by Carl Sassenrath
* How to Handle User Interface Events, by Rebol Technologies
* The whole video tutorial, by Nick Antonaccio
* Introducing Rebol with Amazingly Easy GUI Programming, by Gregg Irwin
* Rebol Primer, by Viktor Pavlu
* Rebol Essentials, by Viktor Pavlu
* Rebol Tutorial
# Rebol Programming/Rebol2 money datatype

Every Rebol2 money value consist of two parts: currency code (can be empty) and a numeric value of the decimal! type.
The parts can be obtained as follows:
# Rebol Programming/Third party

Brief descriptions of third party and other tools commonly used

## See also

* Rebol Programming/Language Features/Database
# Rebol Programming/Language features
## Language Features

* Types
* Series
* Control Structures
* Objects
* Parse
* Dialects
* Recursion
* Database
* View
* VID
* Rebcode
# Rebol Programming/Money datatype

Rebol2 and Rebol3 have different implementation of this datatype:
* Rebol2 money datatype
* Rebol3 money datatype
# Pinyin/Whale

See also
# Pinyin/Where Is The Wind

Search wind
Where is the wind?
Where is the wind? Wind is above grassland, because grass bends down.
Where is the wind? Wind is above flowers, because flowers nod.
Where is the wind? Wind is above willow branches, because willow branches swing.
Where is the wind? Wind is above the sea, because the waves come.
feng zaih naalii?
feng zaih naalii? feng zaih caaodih shahng, yinweih caao wanyao le.
feng zaih naalii? feng zaih huaduoo shahng, yinweih huaduoo diaantour le.
feng zaih naalii? feng zaih liuuzhi shahng, yinweih liuuzhi baaidohng le.
feng zaih naalii? feng zaih dahhaai shahng, yinweih haailahng lair le.
# Pinyin/While

See also
# Pinyin/White

See also
# Pinyin/What Is Time?

Search time
What is time?
Time is shapeless, tasteless, soundless.
When we eat, when we sleep, it flows away quietly.
Time is like a magician, always unwittingly turns humans into, a head full of white hair, a face full of wrinkles.
Time will never come back once flown away, where has time gone? What is time?
shirjian daohdii shih shern'me?
shirjian shih wurxirng d, meiryoou weihdaoh d, meiryoou shengyin d.
zaih woo'men chifahn shir, zaih woo'men shuihjiaoh shir, ta jiuh qiaoqiao dy liurshi le.
shirjian haaoxiahng 1'g morshuhshi, zoongshih bujingyih dy baa rern biahnde, maantour bairfah, maanliaan zhouhwern,
shirjian guorquh le jiuh bu huirlair, shirjian dou quh naa'r le? shirjian daohdii shih shern'me?
# Pinyin/When

See also
# Pinyin/Wheat

See also
Wheat is a grain that people eat. It is a kind of grass seeds. Wheat is a staple food. It is originated from the Middle East. Now it is cultivated worldwide.
xiaaomaih shì rénmen chī de gǔwù. Tā shì yīlèi cǎo de zhǒngzǐ. Xiǎomài shì yīzhǒng zhǔyào shíliáng. Tā yuánzì Zhōngdōng, xiànzài quánqiú zhòngzhí.
# Pinyin/Wheel

A wheel is a round object that turns around and around to make vehicles move.
lurn'zi shih yuarnxirng d wuhtii, xuarnzhuaan shir jiaotong gongjuh yirdohng.
# Pinyin/Which

See also
# Quantum Chemistry/Example 17

The force constant of HCl is determined computationally to be 480 N/m. Given this information find the frequency of EM radiation required to excite the HCl molecule from its ground state to its first excited state.

## Solution

Given the bond constant (k) of HCl , we use the relationship between fundamental frequency and bond constant to find the bond constant.
$\nu _{0}={\frac {1}{2\pi }}{\left({\frac {k}{\mu }}\right)}^{1/2}$
where $K$ is the bond constant.
          $\mu$ is the reduced mass of HCl.
To find the reduced mass of HCl the masses of H and Cl are multiplied and divided by the sum of the masses.
$\mu ={\frac {m_{1}\cdot m_{2}}{m_{1}+m_{2}}}$
For HCl the reduced mass is calculated as
$\mu ={\frac {1.007842u\cdot 35.453u}{1.007842u+35.453u}}=0.979982u$
convert to the SI unit of Kg
$1u=1.66054\cdot 10^{-27}Kg$
$0.979983u=0.979983u\cdot 1.66054\cdot 10^{-27}Kg$
$\mu =1.6273\cdot 10^{-27}Kg$
To find the fundamental frequency
$\nu _{0}={\frac {1}{2\pi }}\left({\frac {K}{\mu }}\right)^{1/2}$
    $={\frac {1}{2\pi }}\left({\frac {480N/m}{1.6273\cdot 10^{-27}Kg}}\right)^{1/2}$
    $=8.6438\cdot 10^{13}hz$
After finding the fundamental frequency, the Energy at different quantum levels can be found by
$E_{v}=h\nu _{0}\left(v+{\frac {1}{2}}\right)$
For the ground state i.e. $v=0$
$E_{0}=h\nu _{0}\left(0+{\frac {1}{2}}\right)$
$E_{0}=h\nu _{0}\left({\frac {1}{2}}\right)$
For the first excited state i.e. $v=1$
$E_{1}=h\nu _{0}\left(1+{\frac {1}{2}}\right)$
$E_{1}=h\nu _{0}\left({\frac {3}{2}}\right)$
The difference in energy between the two states is
$\Delta E=E_{1}-E_{0}$
$\Delta E=h\nu _{0}\left({\frac {3}{2}}\right)-h\nu _{0}\left({\frac {1}{2}}\right)$
$\Delta E=h\nu _{0}$
and Energy is defined as Planck's constant multiplied by frequency
$\Delta E=hv$
$hv=h\nu _{0}$
$v=\nu _{0}=8.6438\cdot 10^{13}hz$
# Quantum Chemistry/Example 14

Show using calculus the most probable position of a quantum harmonic oscillator in the ground state (n=0)
Question:
What is the most probable position of a quantum harmonic oscillator at the ground state? Calculate this using the probability density equation to find the most probable position at n=0.
    Probability distribution $P_{n}(x)=N_{n}^{2}H_{n}({\sqrt {\alpha }}x)^{2}e^{-\alpha x^{2}}$
Solution:
The Hermite polynomial at n=0 is:
$H_{0}(x)=1$
The normalization factor at n=0 is:
$N_{0}={\frac {1}{\sqrt {2^{1}1!}}}\left({\frac {\alpha }{\pi }}\right)^{1/4}=\left({\frac {\alpha }{\pi }}\right)^{1/4}$
α is a constant and is equal to:
$\alpha ={\sqrt {\frac {k\mu }{\hbar }}}$
The probability distribution at n=0:
$P_{n=0}(x)=\left({\frac {\alpha }{\pi }}\right)^{1/2}e^{-\alpha x^{2}}$
The most probable position is when the maximum probability distribution is:
${\frac {\partial P}{\partial x}}=0$
Applying this partial derivative to the probability distribution gives:
${\frac {\partial }{\partial x}}\left({\frac {\alpha }{\pi }}\right)^{1/2}e^{-\alpha x^{2}}=0$
The constants can be taken out of the derivative:
$\left({\frac {\alpha }{\pi }}\right)^{1/2}{\frac {\partial }{\partial x}}e^{-\alpha x^{2}}=0$
The derivative gives:
$\left({\frac {\alpha }{\pi }}\right)^{1/2}[-2\alpha xe^{-\alpha x^{2}}]=0$
Since it is equal to zero the constants can be divided out leaving:
$[-2\alpha xe^{-\alpha x^{2}}]=0$
Since all of the parts are multiplied they can be divided out leaving:
$x=0$
The point where the probability distribution is at a maximum for the ground state of n=0 for the quantum harmonic oscillator is 0.
# Probability and Statistics
## Probability

* Introduction to probability
** Lecture 1
** Lecture 2
** Lecture 3
** Lecture 4
* Random variables and distributions

## Statistics

* Sampling distributions
* Inferences of means and variances
* Inferences concerning proportion, regression analysis
* Non-parametic tests and Bayesian statistics
# Probability Theory

Probability theory is the study of reasoning with incomplete information. The laws of logic govern all correct reasoning when operating under conditions of perfect information. If$A$ is true, and if $A$ implies $B$, then we may deduce that $B$ is true as well. But in many cases, we may be uncertain about whether $A$ is true or not. Probability theory governs all correct reasoning when operating under conditions of incomplete or unreliable information. As such, it is an extremely useful field of study, with many applications.

### Fundamental concepts

* Probability spaces
* Conditional probability
* Independence
* Random variables

### Probabilities on finite sets

* Finite probability spaces
* Random variables on finite probability spaces
* Sums of independent random variables on finite probability spaces

### Partition Functions
## Sources

* von Mises, Richard (1964). Mathematical Theory of Probability and Statistics. New York and London: Academic Press.
* Kolmogorov, Andrey (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung. Berlin: Springer.
* Itô, Kiyosi (1984). Introduction to probability theory. Cambridge u.a., Univ. Pr.
* Kallenberg, Olav (1997). Foundations of modern probability. New York: Springer.
* Loève, Michel (1963). Probability Theory I. D. van Nostrand.
# Probability Theory/Independence

Definition (independence of events):
Let $(\Omega ,{\mathcal {F}},P)$ be a probability space and let $A,B\in {\mathcal {F}}$. $A$ and $B$ are said to be independent iff $P(A\cap B)=P(A)P(B)$.
Remark (independence and conditional probability):
Using the definition of conditional probability, if e.g. $P(A)\neq 0$ we may rephrase the independence of $A$ and $B$ as $P(B)=P(B|A)$.
# Probability Theory/Kolmogorov and modern axioms and their meaning
## Fundamental definition

Definition 2.1 (Kolmogorov's axioms):
Let $\Omega$ be a set, and let ${\mathcal {F}}$ be an algebra of subsets of $\Omega$. Let furthermore $P:{\mathcal {F}}\to [0,1]$ be a function satisfying
* $P(\Omega )=1$ and
* $P(A+B)=P(A)+P(B)$.
Then the triple $(\Omega ,{\mathcal {F}},P)$ is called a probability space.
Note in particular that
    $P(\emptyset )=0$,
since $P(\Omega )=P(\Omega +\emptyset )=P(\Omega )+P(\emptyset )$.
Note that often probability spaces are defined such that the algebra of subsets is a sigma-algebra. We shall revisit these concept later, and restrict ourselves to the above definition, which seems to capture the intuitive concept of probability quite well.

## Elementary theorems

In the following, $(\Omega ,{\mathcal {F}},P)$ shall always be a probability space.
Lemma 2.2:
For $A\in {\mathcal {F}}$,
    $P(A)+P(A^{c})=1$.
Lemma 2.3:
For $A_{1},\ldots ,A_{n}\in {\mathcal {F}}$,
    $P\left(\sum _{j=1}^{n}A_{j}\right)=\sum _{j=1}^{n}P(A_{j})$.
Lemma 2.4:
For $A,B\in {\mathcal {F}}$,
    $P(A\cup B)=P(A)+P(B)-P(AB)$.

### Exercises

* Exercise 2.2.1: Prove lemmas 2.2-2.4.