MATHHX B

MATHHX B

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12.2 Endelige sandsynlighedsfelter

Dette kapitel er for elever på 2024-ordningen. Dvs. elever startet i 2024 eller senere. Er du på den gamle ordning skal du i stedet læse dette kapitel.

Udgangspunktet for sandsynlighedsregning er noget tilfældigt som f.eks. et terningkast. Vi kalder det tilfældige udgangspunk (terningkastet) for et stokastisk eksperiment.

Øvelse 12.2.1

Stokastiske eksperimenter kan være alt muligt.

  • a) Nævn tre stokastiske eksperimenter.

Løsning 12.2.1

  • a) F.eks., et terningkast, et kast med en mønt, eller når man fisker ænder i Tivoli (men jeg synes aldrig man vinder uhhuhuu).

  • Definition 12.2.1
    For et stokastisk eksperiment definerer vi:

    Udfald

    Et udfald er resultatet af det stokastiske eksperiment. Vi betegner udfaldene med lille \(u\).

    Udfaldsrum

    Et udfaldsrum er mængden bestående af alle udfald. Vi betegner udfaldsrummet med \(U\).

  • Eksempel 12.2.1
    Vi kaster en terning.

    Udfaldene er: \(u_1={\Large ⚀} \), \(u_2={\Large ⚁} \), \(u_3={\Large ⚂} \), \(u_4={\Large ⚃} \), \(u_5={\Large ⚄} \) og \(u_6={\Large ⚅} \)

    Udfaldsrummet er: \(U=\{{\Large ⚀} , {\Large ⚁} , {\Large ⚂} , {\Large ⚃} , {\Large ⚄} , {\Large ⚅} \}\).

Øvelse 12.2.2

Antag at vi kaster en mønt. Bestem:

  • a) Udfaldene

  • b) Udfaldsrummet

Løsning 12.2.2

  • a) Udfald: \(u_1=\textrm {plat}\), \(u_2=\textrm {krone}\).

  • b) Udfaldsrum: \(U=\{\textrm {plat},\textrm {krone}\}\)

Til udfaldene i udfaldsrummet hører sandsynligheder. Man kan godt have uendeligt mange udfald, men det er lidt kompliceret, så i første omgang vil vi begrænse os til endelige udfaldsrum.

  • Definition 12.2.2
    Lad \(U=\{u_1,u_2,\ldots ,u_n\}\) være et endeligt udfaldsrum. Vi definerer:

    Sandsynlighedsfunktion:

    En sandsynlighedsfunktion er en funktion \(P\), som til et hvert udfald \(u\) knytter sandsynligheden for dette udfald \(P(u)\). Funktionen skal opfylde:

    • 1. \(P(u_i)\geq 0\) for alle \(u_i\) i \(U\)

    • 2. \(P(u_1)+P(u_2)+\cdots + P(u_n)=1\)

    Sandsynlighedsfelt:

    Udfaldsrummet \(U\) sammen med sandsynlighedsfunktionen \(P\) kaldes et endeligt sandsynlighedsfelt og betegnes \((U,P)\).

Øvelse 12.2.3

De to krav i definitionen af en sandsynlighedsfunktion udtrykker velkendte egenskaber ved sandsynligheder.

  • a) Hvad betyder kravet \(P(u_i)\geq 0\) for alle \(u_i\) i \(U\)?

  • b) Hvad betyder kravet \(P(u_1)+P(u_2)+\cdots + P(u_n)=1\)?

Løsning 12.2.3

  • a) Sandsynligheden for et udfald er altid et ikke-negativt tal (positivt eller nul). Vi kan ikke have negative sandsynligheder!

  • b) Sandsynlighederne lagt sammen skal give \(100\%\).

Har vi et sandsynlighedsfelt \((U,P)\), vil vi ofte angive sandsynlighedsfunktionen med et sildeben . Vi kalder kalder sildebenet en sandsynlighedstabel.

  • Eksempel 12.2.2
    Sandsynlighedsfunktionen for et terningkast kan beskrives ved følgende sandsynlighedstabel.

    .
    \(u_i\) \({\Large ⚀} \) \({\Large ⚁} \) \({\Large ⚂} \) \({\Large ⚃} \) \({\Large ⚄} \) \({\Large ⚅} \)
    \(P(u_i)\) \(\frac {1}{6}\) \(\frac {1}{6}\) \(\frac {1}{6}\) \(\frac {1}{6}\) \(\frac {1}{6}\) \(\frac {1}{6}\)

Øvelse 12.2.4

Vi kaster en mønt.

  • a) Opskriv en sandsynlighedstabel for sandsynlighedsfeltet.

Løsning 12.2.4

  • a)
    \(\begin {array}{ | c | c | c |} \hline u_i & \text {plat} & \text {krone} \\ \hline P(u_i) & \frac {1}{2} & \frac {1}{2} \\ \hline \end {array}\)

Øvelse 12.2.5

Ved kommunalvalget i 2013 gik det således for sig på Læsø:

.
Socialdemokraterne (A) \(23{,}1\% \)
Læsø Liste \(24{,}4\%\)
Samarbejdslisten \(11{,}6\%\)
Venstre (V) \(16{,}1\%\)
Læsø Borgerliste \(10{,}1\%\)
Dansk Folkeparti (O) \(9{,}2\%\)
Det Konservative Folkeparti (C) \(5{,}5\%\)

Vi tager nu en tilfældig borger på Læsø, som har stemt på et parti til kommunalvalget.

  • a) Opskriv sandsynlighedstabellen, der beskriver denne borgers stemme.

Løsning 12.2.5

  • a)

    .
    \(u_i\) A L. Liste Samarb. V L. Borger O C
    \(P(u_i)\) \(0{,}231\) \(0{,}244\) \(0{,}116\) \(0{,}161\) \(0{,}101\) \(0{,}092\) \(0{,}055\)

Øvelse 12.2.6

Lad \(U=\{u_1,u_2,u_3,u_4\}\) være et udfaldsrum og betragt tabellen:

\(\begin {array}{ | c | c | c | c | c |} \hline u_i & u_1 & u_2 & u_3 & u_4 \\ \hline P(u_i) & 0{,}5 & -0{,}1 & 0{,}4 & 0{,}1 \\ \hline \end {array}\)

Tabellen kan ikke være en sandsynlighedstabel, da \(P\) ikke opfylder kravene i definition 12.2.2.

  • a) Hvilke krav er det \(P\) ikke opfylder?

Løsning 12.2.6

  • a) Begge to. \(P(u_i)\geq 0\) for alle \(u_i\) i \(U\) er ikke opfyldt idet, at \(P(u_2)<0\) og \(P(u_1)+P(u_2)+\cdots + P(u_n)=1\) er ikke opfyldt, da sandsynlighederne ikke giver \(1\) tilsammen. Stakkels \(P\), den ville så gerne være en sandsynlighedsfunktion, men det er den ikke.

Den næste øvelse (og mange af de følgende øvelser) indeholder brøker. Hvis du ikke kan finde ud af brøkregning og ikke orker at lære det, så kan du skrive brøkerne ind i GeoGebra og den vil regne det for dig

Øvelse 12.2.7

En elev laver en snydeterning. Sandsynlighedstabellen for et kast med denne terning ser således ud:

\(\begin {array}{ | c | c | c | c | c |c | c |} \hline u_i & {\Large ⚀} & {\Large ⚁} & {\Large ⚂} & {\Large ⚃} & {\Large ⚄} & {\Large ⚅} \\ \hline P(u_i) & \frac {1}{12} & \frac {1}{6} & \frac {1}{6} & \frac {1}{6} & \frac {1}{6} &\text {?}\\ \hline \end {array}\)

  • a) Bestem det manglende felt i sandsynlighedstabellen.

Løsning 12.2.7

  • a) Der skal stå \(\frac {1}{4}\).

Ekstra

Kan man finde ud af at bruge summationstegn, kan definitionen af et sandsynlighedsfelt udtrykkes mere klart:

  • Definition 12.2.3
    Lad \(U\) være et endeligt udfaldsrum. Vi definerer:

    Sandsynlighedsfunktion:

    En sandsynlighedsfunktion er en funktion \(P\) som til et hvert udfald \(u\) knytter sandsynligheden for dette udfald \(P(u)\). Funktionen skal opfylde:

    • 1. \(P(u)\geq 0\) for alle \(u\in U\)

    • 2. \(\sum _{u\in U}P(u)=1\)

    Sandsynlighedsfelt:

    Udfaldsrummet \(U\) sammen med sandsynlighedsfunktionen \(P\) kaldes et endeligt sandsynlighedsfelt og betegnes \((U,P)\).

Skrivemåden \(\sum _{u\in U}P(u)\) betyder at vi skal tage alle udfaldene \(u\) i \(U\) og lægge deres sandsynligheder sammen. Vi husker at \(\in \) betyder ”tilhører” (i denne sammenhæng vil man læses det som ”i” eller ”tilhørende”).