Verified answer

You have 3 characters and 5 tools. So that's a total of 15 possible ways to pick a character and tool.

You asked for the probability that you "won't be a hobbit or use an umbrella".

As I interpret that, you only want to count pairs where you are an elf or human (not a hobbit) and you use anything but an umbrella (e.g. magic, sword, shield or slingshot).

You have 2 choices for a desired character and 4 choices for tool that isn't an umbrella. So that is 8 ways you could get a desirable pairing.

The probability is simply the ratio of favorable outcomes (8) to total possible outcomes (15).

Answer:

8/15

UPDATE:

My interpretation of your English phrase was as:

not(hobbit or umbrella)

That's equivalent to saying:

(not hobbit) and (not umbrella)

There are 2 non-hobbits characters and 4 non-umbrella tools = 8 pairings

Your phrase could also be interpreted as:

(not hobbit) or (umbrella)

In other words, you are okay if the character is not a hobbit (regardless of what tool they use) or you are okay with any character as long as they use an umbrella. I think that's how llaffer interpreted it and that's why he gets 11 favorable outcomes. 

If a Hobbit defends himself against an Orc with an umbrella He doesn't need statistics, He needs a casket.

The probability of two ORed events (I'll call A and B) is:

P(A or B) = P(A) + P(B) - P(A and B)

An example of this is to look at a deck of cards.  The probability of a 5 or a heart.  Adding the probability of a 5 and the probability of a heart adds the 5 of hearts twice so you have to subtract it out (the and).

So your probability is:

P(not a hobbit or use an umbrella) = P(not a hobbit) + P(use an umbrella) - P(not a hobbit and use an umbrella)

P(not a hobbit) = 2/3

P(use an umbrella) = 1/5

P(both) = 2/3 * 1/5 = 2/15

So the OR probability is:

P(not a hobbit or use an umbrella) = 2/3 + 1/5 - 2/15

P(not a hobbit or use an umbrella) = 10/15 + 3/15 - 2/15

P(not a hobbit or use an umbrella) = 11/15