Question

In: Statistics and Probability

Show that, for any events E and F, P(E ∪ F) = P(E) + P(F) −...

Show that, for any events E and F, P(E ∪ F) = P(E) + P(F) − P(E ∩ F). Only use the probability axioms and indicate which axiom you use where

Expert Solution

Solution :

This is the additional law of probability theorem .

P(E ∪ F) = P(E) + P(F) − P(E ∩ F).

Let E and F be the two events on sample space .

E U F can be expressed as the union of two disjoint events E and E^' F

Therefore,

E U F = E U (E' F)

P(E U F) = P(E U (E' F)) = P(E) + P(E' F) ....................(1)

The event F can be expressed as a union of two disjoint events (E F ) and (E' F).

B = (E F) U (E' F)

P(B) = P(E F) U P(E' F)

We know that,

P(E' F) = P(F) - P(E F) ....................(2)

Plug the value of expression (2) in expression (1) we get,

P(E ∪ F) = P(E) + P(F) − P(E ∩ F)..

orchestra answered 2 years ago

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