� Tree diagram : 1 . Path I : P ( A�B)
� pq
2 . Path I + Path III : �
P ( B) � P ( A� B) � P ( A�
� B)
� pq � ( 1 � p) r
� Bayes ’ theorem : 1 .
2 .
P ( A) P ( B | A) P ( A| B)
� for two events
P ( A) P ( B | A) � P ( A�) P ( B | A�) P ( Ai) P ( B | Ai)
P ( Ai | B) �
( i �1 , 2 , 3 )
P ( A ) P ( B | A ) �P ( A ) P ( B | A ) �P ( A ) P ( B | A ) three events
1 1 2 2 3 3 for
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Discrete Probability Distributions
� Properties of a discrete random variable X :
X x
1 x
2 … x n
P ( X x)
� P ( X � x1
) X x2 1 . P ( X � x1) � P ( X � x2) � � P ( X � x n
) � 1
P ( � ) … P ( X � x n
)
2 . E ( X ) � x1P ( X � x1 ) � x2P ( X � x2) � � x P ( X � x ): Expected value of X 3 . E ( X ) � 0 if a fair game is considered n n
� Properties of a discrete random variable X :
1 . 2 .
X x
1 x
2 … x n
P ( X x)
� P ( X � x1
) X x2
2 2 2 2 1 1 2 2 n
P ( � ) … P ( X � x n
)
E ( X ) � x P ( X � x ) � x P ( X � x ) � � x P ( X � x ) Var ( X ) E ( X ) ( E ( X ))
2 2
� � : Variance of X
n
� Linear transformation of a random variable X : 1 . E ( aX �b) � aE ( X ) � b : Expected value of X 2 .
2
Var ( aX �b) � a Var ( X )
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