Probability(1)YanHuang

Definition

• Afiniteprobabilityspaceisdenotedby(", $) where- ! isafiniteset(thesamplespace),and- " isafunction! → [0,1] (theprobabilitymeasure)suchthat

) "(+)�

.∈0= 1

Wheneverhearing“probability”,makesurethatyouareclearwhattheprobabilityspaceis: whatisthesamplespaceandwhatistheprobabilitymeasureonit.

ProbabilitySpace(S, P )• Anoutcomeisapointin".• Anevent isasubsetof".

UniformDistribution

• Everypointin" isequiprobable

$(&) =1"

Probabilitiesofevents

• Let) beaneventofprobabilityspace(", $).Theprobabilityofevent) is

$()) ≔ ∑ $(&)�%∈' .

Examples

• Tossafairdicetwice,whatistheprobabilitythatthetwooutcomesaddupto5?

Examples

• Tossafairdicethreetimes,whatistheprobabilitythatthesumoftheoutcomesislessthan10?

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?Straightflushisapokerhandcontainingfivecardsofsequentialrank,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 pokerhandcontainingfourcardsofthesamerankandonecardofanotherrank,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?Afullhouseisapokerhandcontainingthreecardsofonerankandtwocardsofanotherrank,suchas3� 3� 3� 6� 6�.

Examples

• Drawing5cardsfromastandarddeckof52pokercards(Foursuits:clubs,spades,diamonds,hearts.Eachsuithasthirteencards:A,2,3,…,10,J,Q,K).- WhatistheprobabilityofthefivecardsbeingaFlush?Aflushisapokerhandcontainingfivecardsallofthesamesuit,butnotallofsequentialrank,suchasK� 10� 7� 6� 4�.

Examples

• Drawing5cardsfromastandarddeckof52pokercards(Foursuits:clubs,spades,diamonds,hearts.Eachsuithasthirteencards:A,2,3,…,10,J,Q,K).- WhatistheprobabilityofthefivecardsbeingaStraight?Astraightisapokerhandcontainingfivecardsofsequentialrank,notallofthesamesuit,suchas7� 6� 5� 4� 3�.