I've been reading this very interesting blog post entitled "A review of probability theory" from Terence Tao. Here are a few quotes from the blog post:

Elements of the sample space $\Omega$ will be denoted $\omega$. However, for reasons that will be explained shortly, we will try to avoid actually referring to such elements unless absolutely required to.


In order to have the freedom to perform extensions every time we need to introduce a new source of randomness, we will try to adhere to the following important dogma: probability theory is only “allowed” to study concepts and perform operations which are preserved with respect to extension of the underlying sample space. (This is analogous to how differential geometry is only “allowed” to study concepts and perform operations that are preserved with respect to coordinate change, or how graph theory is only “allowed” to study concepts and perform operations that are preserved with respect to relabeling of the vertices, etc..) As long as one is adhering strictly to this dogma, one can insert as many new sources of randomness (or reorganise existing sources of randomness) as one pleases; but if one deviates from this dogma and uses specific properties of a single sample space, then one has left the category of probability theory and must now take care when doing any subsequent operation that could alter that sample space. This dogma is an important aspect of the probabilistic way of thinking, much as the insistence on studying concepts and performing operations that are invariant with respect to coordinate changes or other symmetries is an important aspect of the modern geometric way of thinking. With this probabilistic viewpoint, we shall soon see the sample space essentially disappear from view altogether, after a few foundational issues are dispensed with.

Let’s give some simple examples of what is and what is not a probabilistic concept or operation. The probability $P(E)$ of an event is a probabilistic concept; it is preserved under extensions. Similarly, boolean operations on events such as union, intersection, and complement are also preserved under extensions and are thus also probabilistic operations. The emptiness or non-emptiness of an event $E$ is also probabilistic, as is the equality or non-equality of two events $E,F$ (note how it was important here that we demanded the map $\pi$ to be surjective in the definition of an extension). On the other hand, the cardinality of an event is not a probabilistic concept; for instance, the event that the roll of a given die gives $4$ has cardinality one in the sample space $\{1,\ldots,6\}$, but has cardinality six in the sample space $\{1,\ldots,6\} \times \{1,\ldots,6\}$ when the values of an additional die are used to extend the sample space. Thus, in the probabilistic way of thinking, one should avoid thinking about events as having cardinality, except to the extent that they are either empty or non-empty.

[The bold is mine.]

This seems to be a very insightful viewpoint. But, in introductory probability classes, it is very common to compute the probability of an event by counting the number of outcomes in the event, and dividing by the total number of outcomes in the sample space (assuming that all outcomes in the sample space are equally likely). Is this bad form? In such cases, would it be preferable to compute the probabilities using an approach that does not involve counting the number of elements of an event?

Perhaps an anology is that, in linear algebra, we often prefer proofs that don't use coordinates.

  • $\begingroup$ I don’t think counting and dividing by the total possible number of equally likely outcomes is bad form, but then I’m not a probabilist, not to mention not even close to a Fields medalist. $\endgroup$ – Steve Kass Feb 11 '18 at 1:09
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    $\begingroup$ The analogy to using coordinates in linear algebra is not all that compelling. In linear algebra we are dealing with continuous quantities (over real or complex fields), and occasionally with discrete quantities (dimension, rank, multiplicities, etc.) In a probability space we usually know ahead of time whether the probability measure is discrete, continuous, or (rarely) a mixture of both. The point is noted, sometimes merely counting outcomes doesn't give equal probabilities. For example, rolling two fair die gives eleven possible sums, but none of the probabilities involve $1/11$. $\endgroup$ – hardmath Feb 11 '18 at 3:41
  • $\begingroup$ Your caveat assuming that all outcomes in the sample space are equally likely is rather important for counting arguments as, for example, there are twelve possible values that two dice can add up to but they are not equally likely. It also runs into problems where every plausible outcome has a probability of $0$, for example on an unaccountably infinite sample space. $\endgroup$ – Henry Jul 18 '18 at 14:02
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    $\begingroup$ It sounds like Terence Tao is trying to formalize the sort of approach I explain in my answer to this question. $\endgroup$ – Jack M Jul 19 '18 at 15:11

In the 17th century when Pascal and Fermat were working on the Problem of points they used fundamental elementary ideas of Enumerative combinatorics which are important and still valid. In elementary schools we still teach addition and multiplication of counting numbers. Just as the Combinatorial principles of Rule of sum and Rule of product are introduced in probability classes. They are still important and valid. What Terence Tao seems to be referring to is some abstract and theoretical framework very far from what is used in the practice of Statistics where the simple ideas introduced in probability classes can be used to solve concrete problems which come up in real life. I am sure that there is some value in Tao's approach when properly understood but there are obstacles to achieving that proper understanding without first having understood the more naive approach and requires some mathematical maturity.


Tao's example suggests to me that he is coming from a pedagogical and/or philosophical perspective. He appears to be saying that the sample space is important only insofar as it captures the underlying structure of the events, and so functions of the sample space that are not preserved by—are not invariants of—refinements of that space should be avoided.

To make this more concrete, consider his example. The sample space we ordinarily associate with the roll of a single die is $\{1, 2, 3, 4, 5, 6\}$. There is nothing to stop us, however, from adjoining to this the roll of a second die, yielding the expanded sample space $\{1, 2, 3, 4, 5, 6\} \times \{1, 2, 3, 4, 5, 6\}$. The reason we do not ordinarily do so is that it doesn't bear on the roll of the first die. But this is a human decision. The machinery of probability is agnostic to such choices.

Tao points out that the cardinality (in terms of outcomes) of the event "a $4$ is rolled on the (first) die" is $1$ in the original sample space (i.e., $\{4\}$), but $6$ in the expanded sample space (i.e., $\{(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6)\}$). Thus, cardinality is not preserved by the expansion in the sample space. This is in contrast to functions such as the probability measure associated with an event, or disjoint union.

In this case, the example seems trivial because the extension is just a Cartesian product, so although cardinality is not preserved, there is a simple mapping between the cardinality in the two sample spaces with respect to any event concerning only the first die (i.e., multiplication/division by $6$).

We could have made the difference clearer by (for instance) rolling an $n$-sided die for the second roll, where $n$ was the result of the first roll. Now, in the expanded sample space, both "a $4$ is rolled on the first die" and "a $1$ or a $3$ is rolled on the first die" have cardinality $4$, but they obviously have different probabilities.

To be sure, no one is really confused by these examples, because it is clear (once one has a basic grounding in probability) what the probability measure should be for each case. I believe Tao's point is that there arise cases where confusion is more plausible, and in any event, nothing is gained by approaching probability via cardinality that isn't captured more directly and more reliably by the notions of events and probability measure.

To answer your more practical questions: What I think Tao might say is that counting up the sample space can be useful, in conjunction with the principle of indifference, specifically in order to determine a useful probability measure. That is to say, the faces of the die are counted up, and turn out to number $6$. Based on symmetry and experience, we apply the principle of indifference. Together, these observations yield a probability measure that assigns $1/6$ to each face.

Once that is done, however, we are done counting. In order to determine the probability associated with "a $1$ or a $3$ is rolled on the first die," we do not count the outcomes that have $1$ or $3$, and divide by the total number of outcomes. Rather, we find the probability measure associated with that event, which by additivity equals the sum of the measures associated with the rolls of $1$ and $3$. That is to say, we compute

$$ P(\text{$1$ or $3$}) = P(1) + P(3) = \frac16+\frac16 = \frac13 $$

rather than

$$ P(\text{$1$ or $3$}) = \frac{|\{1, 3\}|}{|\{1, 2, 3, 4, 5, 6\}|} = \frac26 = \frac13 $$

One value of this approach is that it emphasizes the primary quality of the probability measure (as compared to cardinality). Admittedly, this is not terribly important when it comes to rolls of a presumed fair die, but it gains significance when we must deal with random variables in general (e.g., a loaded die). It is not an intuitive idea up front, compared with counting up outcomes, but a little extra investment of effort and careful thinking at the start can prevent some misconceptions later on down the road.


I feel counting being on an incomparable level of abstraction than event.

I understand Probabilistic concept as a mathematical object derived from the Probabilistic Space. Namely, that means you have a probability measure $P$ already defined. If you wanted to dispute about cardinality of an event you should have it based on the Probabilistic Space.

On the other hand, counting outcomes is used to define the probability measure, I mean "before" you have the Probabilistic Space.

In fact, it means you are constructing a Probabilistic Space from $\Omega$, already existing (counting) measure $\mu$, $\mu(\Omega) < \infty$ such that:

  1. $F = 2^\Omega$,
  2. $P: A \in F \mapsto \mu(A)/\mu(\Omega)$.
  • $\begingroup$ Too bad I'm not knowledgeable enough to understand this answer, so I am unable to upvote. $\endgroup$ – Craig Hicks Jul 23 '18 at 21:11

Does Tao's rule of form provide human-beneficial insight into the famous two fours problem? A pair of dice is thrown. At least one of them is a four. What is the chance the other one is a four also?

The common misstep is treat the question as equivalent to "if the first die is a four what is the chance of the second being a four?".

By using Tao's rule of form we know we that (unordered) $\{1,\ldots,6\} \times \{1,\ldots,6\}$ cannot be UN-extended into two independent $\{1,\ldots,6\}$ spaces.

This is just the reverse ordering of Brian Tung's answer's second paragraph. The reverse order being illegal, rather than redundant, because the full space is being used.

And neither is it incompatible with Somos' answer because computation of unordered selections is part of the curriculum and the two fours problem is really just an exercise within that curriculum, whether or not Tao's wording is used.


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