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# Class2Notes - 0.1 GENERAL CONSEQUENCES OF PROBABILITY...

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0.1. GENERAL CONSEQUENCES OF PROBABILITY AXIOMS 1 0.1 General Consequences of Probability Axioms Theorem 1 P [ A ] ° 1 ; 8 A 2 S Proof. A + A = S = ) P ° A + A ± = P [ S ] = 1 A A = ° = ) P ° A + A ± = P [ A ] + P ° A ± ) P [ A ] + P ° A ± = 1 = ) P [ A ] = 1 ± P ° A ± Note: P ° A ± ² 0 for any event A , by positivity axiom. ) P [ A ] ° 1 : Theorem 2 P [ A ] ° P [ B ] and P [ B ± A ] = P [ B ] ± P [ A ] if A ³ B; 8 A; B 2 S Proof. (Analogous to previous proof, with set B here playing role of S .) A + ( B ± A ) = B = ) P [ A + ( B ± A )] = P [ B ] A ( B ± A ) = ° = ) P [ A + ( B ± A )] = P [ A ] + P [ B ± A ] ) P [ A ] + P [ B ± A ] = P [ B ] = ) P [ A ] = P [ B ] ± P [ B ± A ] Note: P [ B ± A ] ² 0 for any event B ± A , by positivity axiom. ) P [ A ] ° P [ B ] Also: P [ A ] + P [ B ± A ] = P [ B ] = ) P [ B ± A ] = P [ B ] ± P [ A ] : Theorem 3 P [ A 1 [ A 2 [ ::: [ A Q ] = P [ A 1 ]+ P [ A 2 ]+ ::: + P [ A Q ] if A 1 ; A 2 ; :::; A Q 2 S are any disjoint sets. This is called the °nite-additivity property. Proof. By the additivity axiom: A 1 + A 2 + ::: + A Q = ( A 1 + A 2 + ::: + A Q ° 1 ) + A Q = ) P [ A 1 + A 2 + ::: + A Q ] = P [ A 1 + A 2 + ::: + A Q ° 1 ] + P [ A Q ] Likewise, P [ A 1 + A 2 + ::: + A Q ° 1 ] = P [ A 1 + A 2 + ::: + A Q ° 2 ] + P [ A Q ° 1 ] , so = ) P [ A 1 + A 2 + ::: + A Q ] = P [ A 1 + A 2 + ::: + A Q ° 2 ] + P [ A Q ° 1 ] + P [ A Q ] Continue in like manner by induction until the theorem statement is ob- tained. Theorem 4 P [ A 1 [ A 2 [ ::: [ A Q ] ° P [ A 1 ]+ P [ A 2 ]+ ::: + P [ A Q ] if A 1 ; A 2 ; :::; A Q 2 S are any (not necessarily disjoint) sets. Proof. First show that this holds for Q = 2 by partition of A 1 + A 2 : A 1 + A 2 = A 1 + A 1 A 2 where A 1 A 1 A 2 = ° . ) P [ A 1 + A 2 ] = P [ A 1 ] + P ° A 1 A 2 ± , by the additivity axiom. Note: A 1 A 2 ³ A 2 = ) P ° A 1 A 2 ± ° P [ A 2 ] , by Theorem 2. ) P [ A 1 + A 2 ] ° P [ A 1 ] + P [ A 2 ] Interpreting this last equation generically, it can be applied by induction to A 1 + A 2 + ::: + A Q = ( A 1 + A 2 + ::: + A Q ° 1 ) + A Q

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2 = ) P [ A 1 + A 2 + ::: + A Q ] ° P [ A 1 + A 2 + ::: + A Q ° 1 ] + P [ A Q ] Likewise, P [ A 1 + A 2 + ::: + A Q ° 1 ] ° P [ A 1 + A 2 + ::: + A Q ° 2 ] + P [ A Q ° 1 ] , so P [ A 1 + A 2 + ::: + A Q ] ° P [ A 1 + A 2 + ::: + A Q ° 2 ] + P [ A Q ° 1 ] + P [ A Q ] Continue in like manner by induction until the theorem statement is ob- tained. (The process of induction here is analogous to that in the proof of Theorem 2, with " ° " here playing the role of " = " there.) Theorem 5 P [ A [ B ] = P [ A ] + P [ B ] ± P [ AB ] ; 8 A; B 2 S . Proof. Analogous to the partition in the °rst part of the proof of Theorem 4: P [ A + B ] = P [ A ] + P ° AB ± Note: AB = B ± AB ) P [ A + B ] = P [ A ] + P [ B ± AB ] Note: AB ³ B = ) P [ B ± AB ] = P [ B ] ± P [ AB ] , by application of Theorem 2. Combining the last two results gives P [ A + B ] = P [ A ] + P [ B ] ± P [ AB ] . Example 6 An experiment consists of one throw of a fair die. There are six possible elementary outcomes: ± 1 ; ± 2 ; ± 3 ; ± 4 ; ± 5 ; ± 6 which correspond to the ap- pearance of faces 1 ; 2 ; 3 ; 4 ; 5 ; 6 ; respectively. Hence the sample space is S = f 1 ; 2 ; 3 ; 4 ; 5 ; 6 g = ) M : = j S j = 6 : Assume that each outcome is equally likely: P [ ± i ] = 1 M = 1 6 ; i = 1 ; 2 ; :::; 6 : Now consider (i.e., de°ne) some events ( A; B; C; D ) by certain criteria. 1. A : An odd number is thrown. ) A = f 1 ; 3 ; 5 g = ) n : = j A j = 3 : So P [ A ] = j A j j S j = n M = 3 6 = 1 2 : 2. B : An even number is thrown. ) B = f 2 ; 4 ; 6 g = ) n : = j B j = 3 : So P [ B ] = j B j j S j = n M = 3 6 = 1 2 :
0.1. GENERAL CONSEQUENCES OF PROBABILITY AXIOMS 3 3. A \ B : An odd and even number appears on one throw.

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Class2Notes - 0.1 GENERAL CONSEQUENCES OF PROBABILITY...

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