Probability of Independent Sets: A,B in Sample Space Z

[ParisSpart]

In a sample space Z = {a, b, c} is the probability distribution of the numbers x, y, 0.3 (in that order). What should be they that any {a, b} and {a, c} are independent?

maybe i need a suggestion if the type i want to use down is correcty or not because i stucked...



I know that for A1,A2,An we have this type : P(A1andA2and...andAn)=P(A1)P(A2)...P(An) but how i can use this if i have sets?

 

[Simon Bridge]

You need a to be independent of b and c, but b and c need not be independent of each other?

What is the definition of "independent" in this context? (Hint: in terms of probabilities.)
You have:

P(A1andA2and...andAn)=P(A1)P(A2)...P(An) but how i can use this if i have sets?

... you don't have to apply the relation to whole sets, the A1,A2,... form a set {A1, A2, ...}.

if A and B are independent, then P(A)+P(B)=?

 

[haruspex]

If I'm interpreting this correctly, you need to find x and y such that P[f∈{a,b}|f∈{a,c}] = P[f∈{a,b}]. Is that it?

 

[Simon Bridge]

Well if you are interpreting it correctly - then how would the relation P[f∈{a,b}|f∈{a,c}] = P[f∈{a,b}] possibly hold? After all, the two sets have a member in common.

It boils down to how you read this bit:

What should be they that any {a, b} and {a, c} are independent?​

... which is "they" and what is "independent" of what?

Presumably "they" is x and y.
"{a,b} is independent", then I'd read that as outcome a is independent of outcome b... I don't think that {a,b} can be independent of {a,c} since they both contain a.

I read the question as saying that {a,b,c} may or may not not be independent, but {a,b} and {a,c} are.
Given that P(c)=0.3, what is P(a) and P(b)?

However: you are closer to the course than I am so you may have an extra insight.

 

[haruspex]

Well if you are interpreting it correctly - then how would the relation P[f∈{a,b}|f∈{a,c}] = P[f∈{a,b}] possibly hold? After all, the two sets have a member in common.

There is exactly one setting of x and y that makes it true.

"{a,b} is independent", then I'd read that as outcome a is independent of outcome b.

Aren't a, b and c atomic events? If so, they are therefore mutually exclusive, not independent. In probability space terms, a set of possible outcomes, like {a, b} constitutes a (nonatomic) event, and you want the events {a, b}, {a, c} to be independent.

 

[Simon Bridge]

Because Z is a sample space? That would make sense OK.
I still think the phrasing is sloppy... which is why I'm uncertain about how I was reading the question.