Probability(1)
YanHuang
Definition
• Afiniteprobabilityspaceisdenotedby(", $) where
- ! isafiniteset(thesamplespace),and
- " isafunction! → [0,1] (theprobabilitymeasure)suchthat
) "(+) = 1
.∈0
Wheneverhearing“probability”,makesurethatyouare
clearwhattheprobabilityspaceis: whatisthesample
spaceandwhatistheprobabilitymeasureonit.
ProbabilitySpace(S, P )
• Anoutcomeisapointin".
• Anevent isasubsetof".
UniformDistribution
• Everypointin" isequiprobable
1
$(&) =
"
Probabilitiesofevents
• Let) beaneventofprobabilityspace(", $).Theprobabilityofevent
) is
$()) ≔ ∑%∈' $(&).
Examples
• Tossafairdicetwice,whatistheprobabilitythatthetwooutcomes
addupto5?
Examples
• Tossafairdicethreetimes,whatistheprobabilitythatthesumof
theoutcomesislessthan10?
Examples
• Drawing5cardsfromastandarddeckof52pokercards(Foursuits:
clubs,spades,diamonds,hearts.Eachsuithasthirteencards:A,2,3,
4,5,6,7,8,9,10,J,Q,K).
- WhatistheprobabilityofthefivecardsbeingRoyalFlush(i.e.,same-suit10,
J,Q,K,A)?
Examples
• Drawing5cardsfromastandarddeckof52pokercards(Foursuits:
clubs,spades,diamonds,hearts.Eachsuithasthirteencards:A,2,3,
…,10,J,Q,K).
- WhatistheprobabilityofthefivecardsbeingaStraightFlush?Straightflush
isapokerhandcontainingfivecardsofsequentialrank,allofthesamesuit,
suchasQ J 10 9 8 (a“queen-highstraightflush”),butnotaroyalflush.
Examples
• Drawing5cardsfromastandarddeckof52pokercards(Foursuits:
clubs,spades,diamonds,hearts.Eachsuithasthirteencards:A,2,3,
…,10,J,Q,K).
- WhatistheprobabilityofthefivecardsbeingaFourofaKind?Fourofakind,
alsoknownas quads,isa pokerhandcontainingfourcardsofthesamerank
andonecardofanotherrank,e.g., 9 9 9 9 J ("fourofakind,nines").
Examples
• Drawing5cardsfromastandarddeckof52pokercards(Foursuits:
clubs,spades,diamonds,hearts.Eachsuithasthirteencards:A,2,3,
…,10,J,Q,K).
- WhatistheprobabilityofthefivecardsbeingaFullHouse?Afullhouseisa
pokerhandcontainingthreecardsofonerankandtwocardsofanotherrank,
suchas3 3 3 6 6 .
Examples
• Drawing5cardsfromastandarddeckof52pokercards(Foursuits:
clubs,spades,diamonds,hearts.Eachsuithasthirteencards:A,2,3,
…,10,J,Q,K).
- WhatistheprobabilityofthefivecardsbeingaFlush?Aflushisapokerhand
containingfivecardsallofthesamesuit,butnotallofsequentialrank,such
asK 10 7 6 4 .
Examples
• Drawing5cardsfromastandarddeckof52pokercards(Foursuits:
clubs,spades,diamonds,hearts.Eachsuithasthirteencards:A,2,3,
…,10,J,Q,K).
- WhatistheprobabilityofthefivecardsbeingaStraight?Astraightisapoker
handcontainingfivecardsofsequentialrank,notallofthesamesuit,suchas
7 6 5 4 3 .