# Impossible event vs Ordinary event

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1. Apr 1, 2015

1. The problem statement, all variables and given/known data
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.

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

3. 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.

2. Apr 1, 2015

### 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.

3. Apr 1, 2015

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?

4. Apr 1, 2015

### 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.

5. Apr 1, 2015

### WWGD

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

6. Apr 1, 2015

### haruspex

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).

7. Apr 1, 2015

### 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.

8. Apr 1, 2015

### haruspex

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

9. Apr 5, 2015

### Stephen Tashi

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 modelled using many different probability spaces.