� 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
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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
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Binomial Distribution
� Properties of a random variable X ~ B ( n, p ) following binomial distribution :
1 . Only two outcomes from every independent trial ( Success and failure ) 2 . n : Number of trials 3 . p : Probability of success 4 . X : Number of successes in n trials
� Formulae for binomial distribution :
1 . |
�n� r n�r
P (
X
� r)
�
�
�p
( 1
� p) for 0
�r� n, r �
�r �
|
2 . |
E (
X
)
� np
: Expected value of X
|
3 . Var ( X ) �np( 1 � p) : Variance of X
4 . np( 1 � p)
: Standard deviation of X 5 . P ( X � r) � P ( X � r �1 ) �1� P ( X � r � 1 )
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