Understanding Event Spaces and Probability Domains: Solving Problems A, B, and C

[XodoX]

Homework Statement

I have some problems that I don't know the answer to.

A) Does there exist an event space, i.e., probability domain that has exactly one set?

B) Is there an event space, i.e. probability domain that has exactly three sets? Is there one that has exactly four sets?

C)If D is an event space, i.e., probability domain and each A and B is in D, then A U B is in D.

Homework Equations

The Attempt at a Solution

I think A is yes, B is no, and C I have no clue. The most important is the WHY. Can someone explain to me why it's yes and no? I'm not sure how to explain it.

 

[lanedance]

I think it would be a good start to consider some discrete sets & the potential subsets of events

- first the single element set {1}, what are the potential subsets?
- now the double element set {1,2}, what are the potential subsets?

if you equate each subset with an event, this may help answer a) & b)

for c) consider any element in A U B, it must be in either A or B... (note the As and Bs here correspond to sets & not questions, hence the use of the lower case)

 

[lanedance]

by the way, I'm guessing a little as the questions not exactly clear what the definitions are

 

[HallsofIvy]

I think it would be a good start to consider some discrete sets & the potential subsets of events

- first the single element set {1}, what are the potential subsets?

You might also want to consider the subsets of the empty set, with no elements.

- now the double element set {1,2}, what are the potential subsets?

if you equate each subset with an event, this may help answer a) & b)

for c) consider any element in A U B, it must be in either A or B... (note the As and Bs here correspond to sets & not questions, hence the use of the lower case)

 

[XodoX]

Thanks. Got a bunch of definitions he gave us. I didn't think the definitions would matter. It's still the same math after all. The defenition for probability domain is:

The statement that D is a probability domain means that D is a collection of sets such that each of the following statements are true:

1.There is a set I in D such that if A is in D, then A is a subset of I;

2. There is a set O in D such that if A is in D, then O is a subset of A;

3. If A is in D and B is in D, then A$$\bigcap$$B is in D;

4. If A is in D, then there is a set A^c in D such that A$$\bigcap$$B^c=0 and A$$\bigcup$$A^c = I;

This is VERY confusing to me. I can't answer the questions using this definition. Can't figure it out.

 

[Oxymoron]

1 & 2.

You say "event space" therefore I assume you are referring to the sigma-algebra of the sample space of your Probability Space. If you look up the definition of a sigma-algebra the answer to 1. and 2. becomes quite obvious. Hint: Power sets.

3.

Once again, the definition of a sigma-algebra will make this obvious. Sigma-algebras are imployed when considering Probability Spaces because of their neat ability to be measurable. We like our subsets (events) of probability spaces to be measurable because then we have a well-defined notion of size. This is why a Probability Space is a triple (Probability Measure, Sigma-Algebra, Sample Space) and not just the Sample Space.

Now that you know why probability spaces have associated sigma-algebras, just look up the definition of it - it will have three properties - one of them help you solve 3.