Probability space/ Sigma Algebras intuition

May 2012
13
0
Amsterdam, The Netherlands
I've got 2 questions regarding probability spaces/events/sigma algebras etc.

1. If you got the event {1,2,3}, you could see this is as 1 or 2 or 3, right ? Is this the standard notation for events ?
How would you notate the event 1 and 2 and 3 ?

2. Why would you take the set of events to be smaller than all the subsets of the sample space?
In my book, they are talking about the properties of a sigma algebra field.
And if the set of events are all the subsets of the sample space this makes sense to me.
But I'm like, why would bother about smaller sets.
For example:
If you throw a dice, and you got the sample space {1,2,3,4,5,6}.
I could just only put the zero set and the sample space in my set of events...
But I'm like, why would you only look at those events ?
And why can I look at some smaller sets, like the trivial one, but not at the set of the events {2,4,6} and {1,2,3,4,5,6}?

I mean, I can see that this is not a sigma field. But to me the trivial sigma field is as useless as set with the events {2,4,6} and {1,2,3,4,5,6}.

Anyone here who can help me out a little bit ?
 

Plato

MHF Helper
Aug 2006
22,507
8,664
1. If you got the event {1,2,3}, you could see this is as 1 or 2 or 3, right ? Is this the standard notation for events ?
How would you notate the event 1 and 2 and 3 ?

2. Why would you take the set of events to be smaller than all the subsets of the sample space?
In my book, they are talking about the properties of a sigma algebra field.
And if the set of events are all the subsets of the sample space this makes sense to me.
I have absolutely no idea what #1 could mean. Can you make it clearer?

#2
Do you know the properties of a sigma algebra?
 
May 2012
13
0
Amsterdam, The Netherlands
Sorry, to be confusing, let's put it in an other way:
We are throwing a dice one time. So the sample space is {1,2,3,4,5,6}.
How can I interpret the event {1,2,3}?

About #2, I know the properties. I can proof something is a sigma-algebra etc.
 

Plato

MHF Helper
Aug 2006
22,507
8,664
Sorry, to be confusing, let's put it in an other way:
We are throwing a dice one time. So the sample space is {1,2,3,4,5,6}.
How can I interpret the event {1,2,3}?
I understand that you are Dutch. The singular of dice in English is die.
The event \(\displaystyle \{1,2,3\}\) is tossing a die and getting a number less than four.
Or perhaps, getting a number that is at most three.

Please, I still do not know what you are asking in #2.
 
May 2012
13
0
Amsterdam, The Netherlands
Okay, so I can say that I can interpret the event {1,2,3} here as getting 1 or 2 or 3.
Can I say that the event {a,b,c} means getting a or b or c ?

Trying to answer my own #2 question:

Could I say that you normally always choose the set of events as power set of the sample space.
As you want to assign a probability to every event in the power set of the sample space.

But when the sample space becomes uncountable infinite, than there arise problems if you want to assign a probability to every event in the power set of the sample space.
I don't understand exactly why, but, in some cases you have to take a smaller set, because to some events you cannot assign a probability.
In those cases, you are looking at subsets of the power set of the sample space that is a sigma algebra.

... ? Something like that ?
Still a little bit confused as you can see, but this kind of the answer I got when searching on google.
 
Last edited:

Plato

MHF Helper
Aug 2006
22,507
8,664
Okay, so I can say that I can interpret the event {1,2,3} here as getting 1 or 2 or 3.
Can I say that the event {a,b,c} means getting a or b or c ?
That has no meaning apart from a well defined experiment.
It could mean selecting the first there letters of the alphabet.
Or it could mean selecting three letters from the word "abacus"

Trying to answer my own #2 question:
Could I say that you normally always choose the set of events as power set of the sample space.
As you want to assign a probability to every event in the power set of the sample space.
Here the point is that we want the whole space to be an event with probability 1.
We want the emptyset to be an event with probability 0.
If \(\displaystyle A\) is an event, we want \(\displaystyle A^c\) to be an event with probability \(\displaystyle 1-\mathcal{P}(A)\).
 
May 2012
13
0
Amsterdam, The Netherlands
Okay, I understand that.

Another question if I may:
Is it true that, if you deal with finite sample spaces you normally choose the "domain" of the probability function as the power set of the sample space ?