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CaptainX
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- TL;DR
- 1. Definition
2. Properties of conditional probability
1. Definition
If E and F are two events associated with the same sample space of a random experment, the conditional probability of the event E given that F has occurred, i.e. P(E|F) is given by
P(E|F) = (E∩F)/P(F) (P≠0)
2. Properties of conditional probability
Let E and F be events of sample space S of an experiment, then we have
2.1 Property 1
P(S|F) = P(F|F) = 1
we know that
P(S|F) = P(S∩F)/P(F) = P(F)/P(F) =1
similiarly, P(F|F)= 1
P(F|F) = P(S|F) = 1
2.2 Property 2
If A and B are any two events of a sample space S and F is an event of S s.t. P(F) ≠ 0, then
P((A∪B)|F) = P(A|F) + P(B|F) -P((A∩B)|F)
In particular, if A and B are disjoint events, then
P((A∪B)|F)=P(A|F)+P(B|F)
2.3 Property 3
P(E'|F) = 1 - P(E|F)
Since S=E∪E' and E and E' are disjoint events.
If E and F are two events associated with the same sample space of a random experment, the conditional probability of the event E given that F has occurred, i.e. P(E|F) is given by
P(E|F) = (E∩F)/P(F) (P≠0)
2. Properties of conditional probability
Let E and F be events of sample space S of an experiment, then we have
2.1 Property 1
P(S|F) = P(F|F) = 1
we know that
P(S|F) = P(S∩F)/P(F) = P(F)/P(F) =1
similiarly, P(F|F)= 1
P(F|F) = P(S|F) = 1
2.2 Property 2
If A and B are any two events of a sample space S and F is an event of S s.t. P(F) ≠ 0, then
P((A∪B)|F) = P(A|F) + P(B|F) -P((A∩B)|F)
In particular, if A and B are disjoint events, then
P((A∪B)|F)=P(A|F)+P(B|F)
2.3 Property 3
P(E'|F) = 1 - P(E|F)
Since S=E∪E' and E and E' are disjoint events.