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Unformatted text preview: green balls: 2 among 3: 3 =3
2 Number of outcomes in the event (product rule): 2 · 3 = 6 ORF 245 Prof. Rigollet
Fall 2012 Counting rules Exercise. Three balls are selected at random from the jar
below.
What is the probability of getting exactly one red ball and two
green balls? Probability of the event {1 red, 2 greens}: 3
6
=
28
56 Rule of complement
P (Ac ) = 1 P (A) ORF 245 Prof. Rigollet
Fall 2012 Ac
A Exercise. Three balls are selected at random from the jar below.
What is the probability that at least one ball is black or red?
Deﬁne the event A ={at least one ball is black or red}
Then Ac={The 3 balls are green} 1
We have P (A ) =
56
c So that P (A) = 1 55
P (A ) =
56
c Independence ORF 245 Prof. Rigollet
Fall 2012 Two events A and B are called independent if
knowing one does not affect the other. This happens
when A and B pertain to two independent
experiments.
Examples:
• Rolling 2 dice: A={Die one is ⚂} B={Die two is ⚄}
• Two consecutive hands in poker:
A={Full house in 1st hand} B={two pairs in 2nd hand}
• Sampling randomly two students on campus:
A={1st student is ORFE} B={2nd student is ORFE}
Sampling from small population ⇏ independence.
(See Section 2.4 in the book) Product rule for
independent events ORF 245 Prof. Rigollet
Fall 2012 If two events A and B are independent, then
P (A B ) = P (A) · P (B ) Exercises.
1.Rolling two dice. What is the probability of ⚅⚅?
P(⚅⚅)=P(⚅)P(⚅)= 1 · 1 = 1
66 36 2.Two students are selected at random: mike and jake.
P(ORFE)=60%. What is the probability one is ORFE and the
other one is not?
P(jake is ORFE and mike is not ORFE)=P(ORFE)P(not ORFE)
=.6(1.6)=24% (= P(mike is ORFE and jake is not ORFE) )
P(one is ORFE and one is not)=2*24%=48% (addition rule). Conditional probability ORF 245 Prof. Rigollet
Fall 2012 What if the events A and B are not independent?
That is: the outcome of A affects the outcome of B and viceversa.
Example: Consider the contingency table for 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 Survival is not
independent of
social status! What is the probability that a randomly selected name on the
passenger manifest corresponds to someone in 1st class who
survived?
P(1st class AND survived)? Conditional probability ORF 245 Prof. Rigollet
Fall 2012 We can deﬁne the conditional probability of A
given B by
P (A B )
P (AB ) = P (B ) We can also deﬁne the conditional probability of B
given A by
P (B A)
P (B A) =
P (A) Note that P (B A) = P (AB ) if P (A) = P (B )
In the...
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 Fall '09

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