Page 97 - DMTH404_STATISTICS

Unit 7: Modern Approach to Probability

Theorem 3. Notes

For any two events A and B in a sample space S
P Bg b
P A  d Bi b  P A  Bg

Figure 7.2

Venn Diagram
Proof.
From the Venn diagram, we can write
B b
B b
B d A  i A Bg b g P A  i A Bg
P B  d
or
Since A  d Bi and A B b g are mutually exclusive, we have
P B b g d  Bi b  Bg

P A
P A
+
P Bg b
or P A  d Bi b  P A  Bg .

Similarly, it can be shown that
P Ag b
P A  d Bi b  P A  Bg

Additive Laws
P A b Bg b + P Bg b Bg
P Ag b
P A


Proof.
From the Venn diagram, given above, we can write
A B  Ad A  Bi or P A b B  g P Ad A  Bi
Since A and A  d Bi are mutually exclusive, we can write
P A  b Bg b + P A  Bi
P Ag d

Substituting the value of P A  d Bi from theorem 3, we get
P A b Bg b + P Bg b Bg
P Ag b
P A


Remarks:
1. If A and B are mutually exclusive, i.e., A B   , then according to theorem 1, we have
P Ag b
P A b Bg  0 . The addition rule, in this case, becomes P A b Bg b + P Bg , which is

in conformity with axiom III.
2. The event A B (i.e. A or B) denotes the occurrence of either A or B or both. Alternatively,
it implies the occurrence of at least one of the two events.
3. The event A B (i.e. A and B) is a compound (or joint) event that denotes the simultaneous
occurrence of the two events.
4. Alternatively, the event A B is also denoted by A + B and the event A B by AB.

LOVELY PROFESSIONAL UNIVERSITY 89