# If P(A)=0, is A a null event?

I know that P(null event)=0, but is the reverse true? i.e. if P(A)=0 is A a null event?

I'm not too sure I even understand what a null event is, to be honest. Could anyone give me an example of one?

• Null event is a set of events that has no occurrence in the probability space mentioned. – SmallChess Aug 13 '14 at 3:57
• cross-posting math.stackexchange.com/questions/895772/… – Stéphane Laurent Aug 13 '14 at 12:52
• @Student That is incorrect and scarcely even makes sense. – whuber Aug 13 '14 at 15:21
• @whuber The answer below says null event is a set of impossible events. Impossible events is also the set of events that have no occurrence in the probability space. Don't you agree? – SmallChess Aug 14 '14 at 2:05
• @student Your language suggests this set could be nonempty, which would be incorrect. – whuber Aug 14 '14 at 4:02

First of all, note that the term ''null event'' is not unambiguous: some sources use it in a sense ''an event that has zero probability'', while others understand it as ''empty set (as an event)''. As the first interpretation makes the question a tautology (of course if the definition of null event is that it's probability is zero, then a null event has zero probability and an event of zero probability is a null event), I'll concentrate on the second interpretation.

In the usual measure theoretic formulation of probability, ''event'' is a set of outcomes; an event is realized if the outcome of the experiment is within the set. Impossible event is the empty set $\emptyset$, i.e. under no outcome of the experiment can this event be realized.

The answer to your question is no. Let $X$ be a random variable with uniform distribution on $\left[0,1\right]$ and $A$ be the event $X=0.5$ (or any other real number on $\left[0,1\right]$). This is obviously not a null event (such random variate can take the value of $0.5$) but has the probability of zero (as the distribution is continuous).

Another example might be having infinite number of heads when flipping a fair coin. (''Infinite number'' might be formalized, but I don't want to make the discussion too technical, consider it intuitively.) This can happen (that is: the event pertaining to it is not an empty set), yet, its probability is zero.

• I don't agree that "null event" is typically called the impossible event. The impossible event is the empty set and it can never occur. A null event is an event that is assigned the probability $0$ by the probability measure; it is not necessarily the empty event. For a continuous random variable $X$, the event $\{X = a\}$ is a null event because it has probability $0$. But the model does not deny the possibility that $X$ will take on the value $a$. $X$ might take on value $a$; just not frequently. The relative frequency on $N$ trials is $o(N)$ and converges to $0$ as $N\to\infty$ – Dilip Sarwate Aug 13 '14 at 7:33
• @DilipSarwate : Well, it is a terminological question then; my answer above pertains to considering ''null event'' to be the same as $\emptyset$ (and not ''event of zero probability''), of course. I did a search for probability "null event" on Google, the results were mixed, but most supported my interpretation. Also note that if we agree with your definition (''null event is an event that is assigned the probability 0'') the question gets meaningless. With the rest of your comment, I completely agree, that was what I also tried to outline. – Tamas Ferenci Aug 13 '14 at 7:38
• I agree with @DilipSarwate. Tamas, I think you should add in your answer that some authors consider a null event as an event having null probability. – Stéphane Laurent Aug 13 '14 at 12:54
• I think, after several re-readings, that this answer is correct but the first paragraph does not clearly convey its meaning. I take it that this answer comes down to "if by null event you mean the empty set--which axiomatically is an event in all probability spaces and also axiomatically has zero probability--then, contrary to the speculation in the question, there exist non-null events of probability zero." The discussion in the comments concerns the alternative reading of the term "null event" as being "any event of zero probability," which makes the question a tautology. – whuber Aug 13 '14 at 15:25
• @Stéphane I agree with you. I would therefore like to see the introductory paragraph in this answer improved to make what it is trying to say clear, unambiguous, and relevant. – whuber Aug 13 '14 at 16:07

A probability space is a triple ($\Omega,\Sigma,P)$ where $\Omega$ is the set of outcomes, $\Sigma$ a sigma algebra on that set ( i.e. set of subsets of $\Omega$ with particular properties) and P is a probability measure on $\Sigma$.

One of the properties of $\Sigma$ is that it contains the empty set $\emptyset$, and the measure of this set must be zero. So the empty set has measure zero $P(\emptyset)=0$.

However there exist non-empty sets that also have a zero measure. E.g for the normal random variable the measure of a singleton ( which is obviously not empty) {a} is $\int_a^a \varphi(x)dx=0$.

Consequently, the measure of a union of singletons, being the sum of the measures of each singleton in the union, is also zero.

So the null event is the empty set that is element of $\Sigma$ and $P(\emptyset)=0$, but $\Sigma$ can contain non-empty sets of probability zero.

This gives rise to concepts like ''almost everywhere''; a property holds almost everywhere if it holds everywhere except on a set that has probability zero.

The idea of a Null event is used to emulate the idea of a failed experiment.

Let's consider the simplistic analogy of flipping a coin. You have four possible outcomes.

First you have the probability that you did, in fact flip a coin. This has a probability of 1 technically speaking.

The second is the Null event (usually denoted with a probability of 0). This null event means the experiment failed. Because you know you flipped a coin with a probability of 1 the only real null events for this experiment involve not being able to read the coin. Maybe you flipped it and it rolled into a crack in the floor, and you were unable to get conclusive results. The null event is not a number used in the math, but models the finite improbability that you cannot complete the experiment. You can think of it as 0+ (if you are familiar with limits/asymptotic expressions) because there aren't a lot of absolutes in statistical analysis.

Third and fourth are the probability that you flipped heads or tails. Both of these are considered to be .5 or 1/2 because there are only two real options.

The null event and the first are used to tell whether the experiment worked or not. Something with a probability of 0 is not necessarily a null event. Take for example the probability that you will draw a red marble from a bag of blue marbles. This has a probability of 0, but is not a null event because you did in fact draw a marble.

Hopefully that clarifies the Null event for you.

• I apologize for the grammatical errors in my response. I am not an English teacher for a reason. – Kalavin Alderac Jan 17 '17 at 2:53
• This doesn't seem to relate to the topic. – Michael R. Chernick Jan 17 '17 at 3:13
• This answer does not seem to agree with standard definitions in probability. Could you supply an authoritative reference? – whuber Jan 17 '17 at 4:04