Page 98 - DMTH404_STATISTICS

Statistics

Notes Corollaries:
From the Venn diagram, we can write P A  b P A  d Bi , where P A  d
1
1. Bg   Bi is the
probability that none of the events A and B occur simultaneously.
b Bi
d 
B
2. P exactly one of A and occursg  P A Bi d  A 
 d
 P A Bi d Bi éSince ( ÇA B ) ( ÇA B ) = ù û
+ P A 
Ç
ë
 P A Bg
 P A Bg b
 P A b g b  + P Bg b  (using theorem 3)
 P A Bg b  (using theorem 4)
 P A Bg
 b
= P(at least one of the two events occur) – P(the two events occur jointly)
3. The addition theorem can be generalised for more than two events. If A, B and C are three
events of a sample space S, then the probability of occurrence of at least one of them is
given by
 b P Ab + P B Cg B Cg



P A B Cg  B Cg  P A b g b   P A b 
A Cg

+ P B Cg b 
 P A b g b   P A Bg b 
Applying theorem 4 on the second and third term, we get
+ P Bg b
B Cg
 P A b g b + P Cg b  Bg b   P B Cg b   .... (1)
 P A Cg b 
+ P A
 P A
Alternatively, the probability of occurrence of at least one of the three events can also be
written as
 b P A  d B  i

P A B Cg   C .... (2)
1
If A, B and C are mutually exclusive, then equation (1) can be written as
 b
P Ag b
P Cg
P A B Cg b + P Bg b .... (3)

+

If A , A , ...... A are n events of a sample space S, the respective equations (1), (2) and (3) can
1 2 n
be modified as
1 b
i d
i d
P A  A ...  A g  å P A b g  å å P A  i + å å å P A  A  i
A
A
2
n
k
j
i
j
n
+ 1 b g b P A  A  ...  A n g ( i   k, etc. ) .... (4)
j

2
1
1 b 1 d ni
P A  A  ...  A g  1 P A  A  ...  A .... (5)
2
2
n
1 b
n
P A  A  ...  A g  å P A b g .... (6)
i
2
n
i 1

(if the events are mutually exclusive)
4. The probability of occurrence of at least two of the three events can be written as
P A B  g b B C  g b A Cg  P A Bg b  + P A Cg 
P B Cg b
 b
 b



+
 b
3P A B Cg b  
P A B Cg

+
 P A B Cg
+ P B Cg b
 b
 P A Bg b  + P A Cg 2  b 

5. The probability of occurrence of exactly two of the three events can be written as
 b
 d
P A B C  i d A B  i A  B Ci  P A B  g b B C  g b A Cg
C d





 b

P A B Cg (using corollary 2)
= P(occurrence of at least two events) – P(joint occurrence of three events)
 P A Bg b  + P A Cg 3  b 
 b
+ P B Cg b

 P A B Cg (using corollary 4)
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