Probability of Events with Complement

Question

Let E and F be events with P(E) = .3, P(F) = .6, and P(E ∪ F) = .7. Find

(a) P(E ∩ F) (b) P(E|F) (c) P(F|E) (d) P(E^c ∩ F) (e) P(E^c|F)

COMMENT: for (a) I got .2 since P(E ∪ F)= P(E) + P(F) - P(E ∩ F). so .3 plus .6 equals .9, and .9-.7=.2.

(b) P(E|F)= P(E ∩ F)/P(F), so .2/.4=.5

For the rest I am sorta confused.

Answer 1

Clearly (c) and (e) are solved similarly to (b).

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Now (d) is solved through the Law of Total Probability.   For any two events $X,Y$, it will be that: $$\mathsf P(X)=\mathsf P(X\cap Y)+\mathsf P(X\cap Y^\complement)$$