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Unformatted text preview: Titanic example we still need to know either
P(1st Class  survived) or P(Survived  ﬁrst class). ORF 245 Prof. Rigollet
Fall 2012 Properties •If A and B are independent: P (A B )
P (A)P (B )
P (AB ) =
=
= P (A)
P (B )
P (B ) Indeed, knowing B does not affect the probability of A •For any events A and B (not necessarily independent):
P (A B ) = P (AB )P (B ) •For any A and B we can ﬁnd P (B A) from
P (A B )
P (B )
P (B A) =
= P (AB )
P (A)
P (A) P (AB ) by: ORF 245 Prof. Rigollet
Fall 2012 Properties P( ⦁  C) is also a probability. In other words, it
satisﬁes all the rules of a usual probability. •Addition rule for disjoint events:
P (A B C ) = P (AC ) + P (B C ) •General addition rule: P (A ⇥ B C ) = P (AC ) + P (B C ) •Rule of complement: P (Ac C ) = 1 P (A ⇤ B C ) P (AC ) •Multiplication rule for independent events:
P (A B C ) = P (AC )P (B C ) ORF 245 Prof. Rigollet
Fall 2012 Back on the Titanic
Survived
No Total 1st class 122 203 325 2nd class 167 118 285 3rd class 528 178 706 crew 673 212 885 Total Class Yes 1,490 711 2,201 P(1st class AND survived)= 203
2201 203
1st class)=325
325 P(survived 
P(1st class)= 2201 P(1st class AND survived)= P(survived  1st class) P(1st class)= 203 325
203
=
325 2201
2201 21 (2008) ORF 245 Prof. Rigollet
Fall 2012 The game show ORF 245 Prof. Rigollet
Fall 2012 Suppose you’re on a game show.You’re given a chance
to choose from 3 different doors. Behind one of the doors is a
behind the other two:
Which door would you choose? The game show ORF 245 Prof. Rigollet
Fall 2012 You’ve chosen door number 1 for example. Next the game show host (who knows what’s behind
each door) opens a door with a goat: say door #3. Is it in your interest to switch to door #2? The game show
To solve the problem, deﬁne the events:
C1: {the car is behind door 1}
C2: {the car is behind door 2}
H3: {the host opens door 3}.
Assume that you ﬁrst chose door 1.
Compute: P(C1) and P(C2  H3). ORF 245 Prof. Rigollet
Fall 2012 The game show ORF 245 Prof. Rigollet
Fall 2012 Random variables ORF 245 Prof. Rigollet
Fall 2012 A random variable X is a variable whose
outcome is random. It is obtained by measuring
the outcome of a random experiment.
Examples:
1. The number on the face of a die
2.The outcome of a basketball game
3. The GPA of a randomly selected (r.s.) student
4. {0,1} that indicates if a drug cures a r.s. patient
5. The median home value of a r.s. US census t...
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 Fall '09

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