SOLUTION: Find P(E) under the assumption that P(E ∪ F) = 0.7 and P(E ∪ F^c ) = 0.9

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Question 1191036: Find P(E) under the assumption that P(E ∪ F) = 0.7 and P(E ∪ F^c
) = 0.9
Answer by ikleyn(52756) About Me  (Show Source):
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Find P(E) under the assumption that P(E ∪ F) = 0.7 and P(E ∪ F^c) = 0.9
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Apply the basic formula of the probability theory for events (sets) A and B of the universal set X

    P(A U B) = (P(A) + P(B) - P(A ∩ B).


You will get

    P(E U F)   = P(E) + P(F)   - P(E ∩ F)      (1)

and

    P(E U F^c) = P(E) + P(F^c) - P(E ∩ F^c)    (2)


Now add these equations (1) and (2).  Substitute the given values  P(E ∪ F) = 0.7   and  P(E ∪ F^c) = 0.9. 

You will get

    0.7 + 0.9 = 2*P(E) + (P(F) + P(F^c)) - (P(E ∩ F) + P(E ∩ F^c)).    (3)


Now take into account that  P(F) + P(F^c) = 1  and  P(E ∩ F) + P(E ∩ F^c) = P(E).


You will get then from (3)

       1.6    = 2*P(E) + 1 - P(E).


Combining like terms, you get the ANSWER :   P(E) = 0.6.

Solved.