Suppose that you visit Shire, and find that 28% of Hobbits believe in the existence of Elves, 16% believe in the existence of Ents, and 9% believe in the existence of both Elves and Ents.

Question

You come across a Hobbit:

i) What is the probability that he will believe in Ents given that he believes in Elves?

ii)Suppose you come across two Hobbits:
What is the probability that they will both believe in Elves? (Assume Independence)

Personal Question: I thought probabilities always add up to 100% or 1, however these add up to 53. Does it not matter? Supposing the question asks, find the probability that the hobbit(assuming you come across one) does not believe in Ents, do you subtract .16 from 1? I just need clarification. Thank you!

Answer 1

conditional probability:
prob (ents given elves)
= prob(endts | elves)
= prob(elves AND ents)/prob (elves)

prob(ents | elves)
= .09/.28 = .3214

prob (2hobbits believing in elves)
= (.28)(.28) = .0794

They do add up to 1 if you add up ALL possible cases, you added up only 3

A good way is to look at the problems with Venn diagrams
Draw 2 intersecting circles , label one elves, the other ents.
put .09 in the intersection of the two.
now look at the "elves" circle, it is supposed to contain .28, but we have already counted .09, so place .19 in the part of the circle not intersecting with the other one.
look at the "ents" circle, it must contain .16 but we already counted .09, leaving .07 for the rest of that circle
Count up the 3 numbers to get .35
That leaves .65 outside of both circles.

So now we have 4 cases
hobbits believing in neither one = .65
hobbits believing in both = .09
hobbits believing in only elves = .09
hobbits believing in only ents = .07
add them up to get 1

When you added .28+.16+.09
wasn't the .09 contained in both .28 and .16 ?
So you added it twice, that's why it made no sense

Answer 2

near the end it should say:

So now we have 4 cases
hobbits believing in neither one = .65
hobbits believing in both = .09
hobbits believing in only elves = .19
hobbits believing in only ents = .07
add them up to get 1