MATHHX B

MATHHX B

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13.2 Endelige sandsynlighedsfelter

Sandsynlighedsregning handler om at regne sandsynligheder for forskellig ting der kan ske. Når der sker noget tilfældigt kalder vi det et stokastisk eksperiment. Det kan f.eks. være kast med en terning eller en mønt.

Øvelse 13.2.1

  • a) Nævn tre stokastiske eksperimenter.

Løsning 13.2.1

  • 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 13.2.1
    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 13.2.1
    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 13.2.2

Antag at vi kaster en mønt. Bestem:

  • a) Udfaldene

  • b) Udfaldsrummet

Løsning 13.2.2

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

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

Til udfaldene i udfaldsrummet hører sandsynligheder. Det er mest simpelt at arbejde med endelige udfaldsrum, så det vil vi begrænse os til i første omgang.

  • Definition 13.2.2
    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. \(P(u)\geq 0\) for alle \(u\in U\)

    • 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 13.2.3

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

  • a) Hvad betyder kravet \(P(u)\geq 0\) for alle \(u\in U\)?

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

Løsning 13.2.3

Forklar hvad de to krav i definitionen af en sandsynlighedsfunktion udtrykker:

  • a) Sandsynligheden for et udfald er altid et ikke-negativt tal (positivt eller nul).

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

Har vi et sandsynlighedsfelt \((U,P)\), vil vi ofte beskrive det med et sildeben for sandsynlighedsfunktionen. Sådan en tabel kalder vi en sandsynlighedstabel.

  • Eksempel 13.2.2
    Sandsynlighedsfunktionen for et terningkast kan beskrives ved følgende sandsynlighedstabel.

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

Øvelse 13.2.4

Vi kaster en mønt.

  • a) Opskriv en sandsynlighedstabel for sandsynlighedsfeltet.

Løsning 13.2.4

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

Øvelse 13.2.5

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 13.2.5

  • a)

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

Øvelse 13.2.6

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

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

Tabellen kan ikke være en sandsynlighedstabel, da \(P\) ikke opfylder kravene i definition 13.2.2.

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

Løsning 13.2.6

  • a) Begge to. \(P(u)\geq 0\) for alle \(u\in U\) er ikke opfyldt i det 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.

Øvelse 13.2.7

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

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

  • a) Bestem sandsynligheden for at slå en 6’er.

Løsning 13.2.7

  • a) Sandsynlighed for at slå 6 med terningen er \(\frac {1}{4}\).

Ekstra

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

  • Definition 13.2.3
    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.