Stat help!! Let E and F be two eventsand P(E)=0.35,P(F)=0.55, and P(E and F compliment)=0.15

a) compute the probability of exactly one of these events occurring
c) P(E or F)=?
d) Are E and F mutually exclusive? Why or why not?
f) P(E compliment | F)

Please explain as well because I hav a test and have no idea what to do. Thanks you so much :)

P(E and ~F) = P(E) - P(E and F)
P(E and F) = P(E) - P(E and ~F) = 0.35 - 0.15 = 0.20

a)

P(exactly 1 event occurring)
= P(E) + P(F) - 2 P(E and F)
= 0.35 + 0.55 - 2*0.20
= 0.50

c)

P(E or F)
= P(E) + P(F) - P(E and F)
= 0.35 + 0.55 - 0.20
= 0.70

d)

No, because mutually exclusive events cannot happen at the same time.
i.e. P(E and F) would be 0 ----> P(E and ~F) would be P(E)

f)

P(~E | F)
= P(~E and F) / P(F)
= (P(F) - P(E and F)) / P(F)
= (0.55 - 0.20) / 0.55
= 0.35 / 0.55
= 7/11
= 0.636363636

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EDIT: A Venn diagram helps

∩ = intersection
∪ = union
~ = not

Draw 2 intersecting circles, label them E and F.
Enter given information:

P(E ∩ ~F) = 0.15
This is the area of circle E that does NOT include the intersection with F:
http://i44.tinypic.com/y04et.png

P(E) = 0.35
This is the sum of all areas that make up circle E
The only missing area is the intersection with F
Since the other region of E = 0.15, then the intersection must = 0.20
http://i39.tinypic.com/2wejn1s.png

P(F) = 0.55
This is the sum of all areas that make up circle F
Since intersection with E = 0.20, then the other missing area must be 0.35
http://i44.tinypic.com/espaw5.png

Now adding all probabilities we get 0.15 + 0.20 + 0.35 = 0.70
So all of region outside of E and/or F has probability of 0.30
http://i43.tinypic.com/9602du.png

Now we can easily answer the questions using Venn diagram:

a) P(exactly one of the events occurring) = 0.15 + 0.35 = 0.50
c) P(E ∪ F) = 0.15 + 0.20 + 0.35 = 0.70

f) P(~E | F):
Since F is a given, then we are only interested in circle F
= 0.20 + 0.35 = 0.55
Now which section of circle F corresponds to ~E ---> 0.35
P(~E | F) = 0.35/0.55 = 7/11

Mαthmφm

Hint: draw a Venn Diagram
a/ question is not complete
c/ answer = 0.7
d/ no
f/ use formula , and Venn diagram