MATHHX B
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#2,,\LWRsiunitxENDTWO }}\)
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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.
-
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.
Øvelse 13.2.3
De to krav i definitionen af en sandsynlighedsfunktion udtrykker velkendte egenskaber ved sandsynligheder.
Løsning 13.2.3
Forklar hvad de to krav i definitionen af en sandsynlighedsfunktion udtrykker:
Har vi et sandsynlighedsfelt \((U,P)\), vil vi ofte beskrive det med et sildeben for sandsynlighedsfunktionen. Sådan en tabel kalder vi en sandsynlighedstabel.
Øvelse 13.2.4
Vi kaster en mønt.
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.
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.
Øvelse 13.2.7
En elev laver en snydeterning. Sandsynlighedstabellen for et kast med denne terning ser således ud:
\(\begin {array}{ | c | c | c | c | c |c | c |} \hline u & {\Large ⚀} & {\Large ⚁} & {\Large ⚂} & {\Large ⚃} & {\Large ⚄} & {\Large ⚅} \\ \hline P(u) & \frac {1}{12} &
\frac {1}{6} & \frac {1}{6} & \frac {1}{6} & \frac {1}{6} &\text {?}\\ \hline \end {array}\)
Ekstra
Kan man finde ud af at bruge summationstegn kan definitionen af et sandsynlighedsfelt udtrykkes mere klart:
Skrivemåden \(\sum _{u\in U}P(u)\) betyder at vi skal tage alle udfaldene \(u\) i \(U\) og lægge deres sandsynligheder sammen.