Impossible event vs Ordinary event

[lahanadar]

Homework Statement

In probability theory,
i) The events A and B are equal with probability 1 iff P(A)=P(B)=P(AB).
ii) The events A and B are equal in probability if only we know P(A)=P(B)
iii) From (i), if an event N equals the impossible event with probability 1, then P(N)=0. Additionally, this does not mean that N={ }

(From Probability, Random Variables and Stochastic Processes, 4th Ed. of Papoulis's book, page:21)

The items from (i) to (iii) are all from the reference given above. My confusions are:

a) For (iii), "if an event N equals the impossible event with probability 1" as defined in (i) then shouldn't that mean P(N)=P({ })=P(N and { })=0 and as a result shouldn't N={ } be in fact correct?

b) I assume (iii) has no issues, and I assume N=/={ }. Then, how is it possible that N can be an event since it does not belong to power set of sample space. From the definition, an event should be choosen from the power set and since N is not impossible event, then it is something does not belong to power set of sample space S.

Homework Equations

Event equality condition: P(A)=P(B)=P(AB)
Power set definition: 2^S
Events are choosen from power set.

The Attempt at a Solution

I think N is impossible event, so we can know it is impossible, plus, that's why we can call it as an event. Correct me if I'm wrong.

 

[WWGD]

What I found confusing is that if the sample space of S is #2^S# , then no event is impossible, in that every event is in the sample space. I would assume an impossible event is one not found in the sample space.

 

[lahanadar]

For rolling a die experiment, possible outcomes are 1 to 6 which forms the sample space. As an example, 7 is not an outcome for this experiment and so not in the sample space. Events for this 6-sided die should be choosen from the power set 2⁶. In this power set, empty set (impossible event) is also an element. From this point of view, impossible event is an event. But since 7 does not belong to any of the members of power set, it is neither event even nor elementary event. Is there any misconception I did here?

 

[haruspex]

It should help to take an actual example: a continuous uniform distribution on (0, 1). The probability of a single point, {x}, 0<x<1, is zero, but it is not impossible.

 

[WWGD]

Which are you referring to? We are looking for impossible events.

 

[haruspex]

Which are you referring to? We are looking for impossible events.

No, you are trying to understand how an event, N, can have zero probability (P(N)=P({})=0), yet not be impossible (N is not empty).

 

[WWGD]

Not me, I am aware of the fact that singletons have measure 0/probability 0 in a continuous distribution; maybe lahanadar does. I am trying to understand what an impossible event is in the OPs layout.

 

[haruspex]

Not me, I am aware of the fact that singletons have measure 0/probability 0 in a continuous distribution; maybe lahanadar does. I am trying to understand what an impossible event is in the OPs layout.

It refers to "the" impossible event, namely, the null set.

 

[Stephen Tashi]

From this point of view, impossible event is an event. But since 7 does not belong to any of the members of power set, it is neither event even nor elementary event. Is there any misconception I did here?

You are correct that {7} is neither and event nor an elementary event in that probability space. In that probability space, it is not technically correct to say that "7 has a a probability of zero" since the mathematical definition of probability space involves an assignment of probabilities only to certain sets. The correct statement is that "7 does not have a defined probability" in that probability space.

The fact that the null set is assigned a probability zero is a consequence of the definition of a probability space. It is required by that definition.

For the same problem ( tossing a fair die) you could model it in a different way by defining the space of elementary outcomes to be all non-negative integers and assigning probabilities of zero to most of them. In that definition of a probability space, it would be correct to say that "the probability of {7} is zero".

A given real life problem doesn't define a unique mathematical model. Usually in a textbook problem, the author expects the exercise to be done with a particular way. If a textbook problem says "What is the probability space for tossing a fair die?", the author probably has in mind a particular space. However, it is not actually correct to ask about "the" probability space for real life situation. Real life situations (including tossing a fair die) can be modeled using many different probability spaces.