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

Question 1191036: Find P(E) under the assumption that P(E ∪ F) = 0.7 and P(E ∪ F^c
) = 0.9
Answer by ikleyn(52780) (Show Source): You can put this solution on YOUR website!
.
Find P(E) under the assumption that P(E ∪ F) = 0.7 and P(E ∪ F^c) = 0.9
~~~~~~~~~~~~~~~

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.

RELATED QUESTIONS
Let P(E) = 0.45, P(F) = 0.55, and P(F ∩ E) = 0.25. Draw a Venn diagram and find the... (answered by ikleyn)

Supppse that E and F are two events and that P(E) = 0.28 and P(F|E) = 0.7 What is P(E and (answered by ikleyn)

E and F are mutually exclusive events. P(E) = 0.15; P(F) = 0.71. Find P(E | F) (answered by ikleyn)

If P(E) = 0.40, P(EorF) = 0.50 , and P(E and F)=0.10 , find... (answered by ikleyn)

if P(E) = 0.40, P(E or F) = 0.50, and P(E and F) = 0.05, find... (answered by ikleyn)

Suppose that E and F are two events and that P(E)=0.4 and P(F|E)=0.5. What is P(E and... (answered by MathLover1)

If E and F are two disjoint events in S with P(E) = 0.6 and P(F) = 0.1, find P(E ∪... (answered by ikleyn)

P(E)=0.35, P(F)=0.35, P(E ∩ F)=0.15 and P((E ∪ F)c)=0.45. Find P(E ∩ FC|F) (answered by ikleyn)

I need to find the equation and solution to this: If P(E) = .4 P(F) = .5 and P( E (answered by Fombitz)