# Example illustrating use of additivity of (probability) measures before introducing independence and conditioning

I would like to be able to illustrate why additivity is a natural property to assume for probability measures. While it is relatively simple to give a short hand waving intuitive analogy to volume or mass I feel this is somewhat unsatisfying, not least since it does not really illustrate how you would use it.

What seems most reasonable to me is to show what amounts to an application of the law of total probability. The problem is that all examples I can think of rely on conditioning or independence, and in the standard measure theoretic (Kolmogorov axiom) formalism the topic of additivity is, by necessity, raised before one can get to independence or conditioning. I would prefer to be able to give at least one motivating example of additivity before going on to the next topic.

The best substitute I have been able to think of is to take a proof of some basic proposition for probability measures that invokes additivity (such as monotonicity with respect to ordering of events) and specialise it to some concrete events. I worry, though, that students with little mathematical background (the ones I'm teaching this semester have not even seen basic calculus) would find such arguments overly contrived or circuitous.

Note that I realise that one could approach the subject completely without considering assignment of probabilities as something reifed as a measure. That being said, since the course is also to cover basic statistics I feel like this is more or less necessary considering how statistics requires one to deal of multiple alternative (sets of) measures corresponding to different hypotheses.

EDIT: In hindsight I think I made the mistake of trying to anticipate responses, thus making the question much more convoluted than necessary. Let me therefore try to state it more concretely.

Is it possible to give an example of how additivity/disjointness can be used to analyse a complex event by partitioning it into simpler parts, before conditioning and/or independence has been introduced. Since the idea is to illustrate how additivity/disjointness encode something important/natural I want an example with some mathematical content (i.e. I would like to avoid a pure appeal to intuition or prototypical examples) but not something purely mathematical (i.e. not only derive additional nice properties measures will have).

The example I ended up using was, if I remember correctly, one that was also hinted at here. Namely that it allows one to motivate the 1/6 probability for a die roll from symmetry of the die. I don't think this worked particularly well though, since I think many people did not even realise this probability assignment was something that had to be justified.

Failing this, can one give some other example (not in terms of partitioning a complex event) achieving the same goal?

• Thank you for this clarification! (I now see what you're going for!) – Pat Devlin Dec 14 '16 at 11:18

Well you run into an issue here.

Namely, (1) you want to say "here's this thing called a probability measure," which is something you're defining for the students. But at the same time (2) you want to say why the probability measure should be additive.

The problem of course is that additivity is a property used to define what a probability measure is in the first place! So they are additive because we define them to be.

So really, you want to argue why this definition is natural. For this, I would suggest a few routes:

(1) this property holds when we have equally likely outcomes, where it amounts to saying $|A \cup B| = |A| + |B|$ when $A$ and $B$ are disjoint.

(2) it is intuitively clear for a lot of people, but the difficulty [if any] is in notation. So if you can get them ok with the notation, they'll be ok with the fact.

(3) You can have them prove some basic stuff like $P(A^c) = 1 - P(A)$ or inclusion/exclusion formula from axioms. This doesn't give intuition so much as practice.

(4) You could have them think from a frequentist perspective. Say "we looked at 20 rocks pulled up from the ocean floor, and here's the data about them [20 row table. Each row has columns for weight, color, and shape]." Then tell the students to predict about what percent of ocean rocks happen to be [gray]. Then ask them to predict what percent are [round]. Then ask them to predict what percent are [blue]. Then predict what percent are [blue or gray]. And predict for [gray or round]. (Et cetera) The idea is that students will all naturally come up with probability for this, and the additive property is clear. It also emphasizes that if the events aren't disjoint then additivity fails. You can get the students to articulate why they used additivity, and why they want their estimates of probability to have this property [even though they don't know the distribution at all].

(5) This is clear if the sample space is finite. In fact, you could just say that $P(A)$ is short-hand for $\sum_{a \in A} P(\{a\}).$

• (1)/(2) still do not illustrate how we use additivity and, arguably, are basically saying "some other notions of measure we have seen satisfy it". (3)/(5) basically boils down to proving that it has the right consequences, which is basically what most of the course is about, I would like to have some good example illustrating additivity before starting to prove that it indeed does give the "right" theory. (4) is more interesting, but would require one to basically start out with a statistical perspective. I feel like there should be some good purely probabilistic example one could take. – tilo.wiklund Dec 13 '16 at 12:33
• I'm not sure what you're looking for. To me, the point is that probability does not start before you define what a probability measure is. Before then, people just have intuition on the thing. So in my opinion, you want to somehow just tap into that intuition. It's sort of boils down to a philosophical question of what should probability be in the first place. Two schools of thought on this are frequentist and bayesian. I discussed how to motivate this from a frequentist perspective, and I'm not quite sure yet how I would do it from a bayesian. – Pat Devlin Dec 13 '16 at 12:58
• When I teach the course, I tell my students these three axioms, and then we just spend 20 or so minutes discussing why they make sense. But the driving force of this discussion is students figuring out for themselves and explaining why they think it makes sense. – Pat Devlin Dec 13 '16 at 13:00
• The idea was not to have it precede the axiomatic definition of measure, but to give an example illustrating why additivity is a reasonable defining property of measures. I'm not so sure people have a good intuitive grasp of this, as I don't think it is totally clear to everyone why disjointness of events is natural to introduce. A way to convey this would be to illustrate that it allows us to analyse the measure of a set by breaking it down into smaller pieces and analyse these separately, but I cannot think of a good way to illustrate this without first introducing independence/conditioning. – tilo.wiklund Dec 13 '16 at 15:56
• I see what you're saying. But ultimately, it seems your hands are somewhat tied in this. You could say (a) "this is a definition of an object called a probability measure, let's accept it as an abstract notion and go from there" [not what you want to do]. Or you could try option (b) saying "there is a notion of probability that you already understood before I gave you this abstract definition. That notion you already know has this property." But (b) is tough if you don't want to accept (1), (4), or (5) above. – Pat Devlin Dec 13 '16 at 16:01

I agree with Pat Devlin answers that the point seems to give an intuition why this part of the definition is natural. I would think that the following example is pretty telling (learned from Tim Gowers) : who will be the replacement to Nico Rosberg as a F1 pilot? Turns out some people gave estimates of probability for several possible pilots, who add up to 5.6 (Fernando Alonso: 6/10 chance; Daniel Ricardo : 6/10 chance; etc.) One interpretation is precisely that these people are unaware that the probability of a union of mutually exclusive events is the sum of their individual probability.

Another example where people get it right: if you ask what is the probability of a regular dice to turn up the face 4, the answer $1/6$ precisely comes from the combination of additivity and equiprobability (which itself can be justified by symmetry).

In particular, we should insist that the core property related to additivity is disjointness (and this cannot be stressed too much).

• I agree here. To my students everything is always additive, and the difficulty is trying to get them to internalize in what sense probability isn't! – Pat Devlin Dec 14 '16 at 11:16
• I definitely agree with your point (especially that one should make sure to stress that the notion of disjointness is important), and as I note in my updated answer I ended up using, basically, your die example. I suspect most students were not really convinced by it though, since many of them did not even understand that one would want to justify the 1/6th probability based on some more elementary principle. – tilo.wiklund Dec 14 '16 at 14:26
• @tilo.wiklund: then you can also use the example of a biased coin (where if you know the probability to get tail, additivity gives you the probability to get heads), or the sum of two dice (but then there is more work to do; the more basic examples of two fair coins with $0$ on one face and $1$ on the other is probably simpler, but also more artificial). Then the values stop being obvious. – Benoît Kloeckner Dec 14 '16 at 19:47
• In both these cases you essentially just end up enumerating basic outcomes again, the nice thing about the die example is that it involves deriving the distribution from a natural (extra-mathematical) assumption of symmetry. – tilo.wiklund Dec 14 '16 at 21:25

Here's another crack at an answer. Let me know if it's closer to what you want.

Bob picks a number from 1 to 100 (uniformly at random). What's the probability that Bob's number doesn't contain the letter "e" when you spell it out?

This has the advantage that {numbers with that property} are hard to understand, but the event is much easier to view as {2 or 4 or 6 or 30 or 32 or ... or 66}. And then the answer is just P(2) + P(4) + ...

I expect students will naturally solve the problem like this, and after they do, you just have to emphasize that what they did was

1. Rewrite the event as a set [skip if this is not an emphasis of the course]
2. Realize that P(this or that) = P(this) + P(that)

Of course, you don't want them to feel like everything is additive. You could perhaps follow this up by asking if P(2 or 4 or 6) = P(2 or 2 or 4 or 6) and letting them discuss this. [at some point, have them argue against the claim that P(2 or 2 or 4 or 6) = P(2) + P(2 or 4 or 6)]

If you really want to emphasize that they're actually ADDING probabilities [as opposed to merely counting], you could somehow make the distribution for each number different. Then they would certainly solve it by adding up P(2) + P(4) + ..., but this is of course harder to swing. Perhaps after they answer for the uniform distribution, you say "Alice also picks a number from 1 to 100, but she does it in a special way. She first flips a coin, and if it lands H, then she will pick the number 2. If it lands T, then she will pick... What's the chance that Alice will pick a number with that property?" or more to the point might be to say "Alice also picks a number from 1 to 100, but she does it in a weird way. 50% of the time, she picks 7, which is her favorite number. 30% of the time, she picks the number 42. 12% of the time, she picks..." [maybe some table for this]

• In fact, I like the set-up "Alice picks a number from 1 to 100 in a weird way; here's a table." You could give the table and then get a lot of mileage out of it. For instance, "from the table, what do you think her favorite number is?" or "what's the chance she picks an even number?" [don't make it remotely close to 50%] or later recall it for random variables or expected value when they mature to it. – Pat Devlin Dec 14 '16 at 11:59
• Sorry for being difficult :) I like this better (the non-uniform one, as you point out the uniform one has similar difficulties to the die example) and I think one could make it work if one finds some way to justify the table of probabilities rather than just giving it. If it just boils down to looking up the individual probabilities in a table you are essentially back at counting. In some way this is what I was trying to do, but failed at justifying the table of probabilities without resorting to arguments in terms of independence and/or conditional probabilities. – tilo.wiklund Dec 14 '16 at 14:23
• I'm glad you like this better (I aim to please). I agree that it would be nice to somehow motivate the entries of the table. You could say that the numbers on the table were found by some sort of experiment, or perhaps by looking at historical data or something. Or you could change it a little bit so it's a weird number generator or something. If you say that it's generated by a computer, then it can have some weird probabilities without making sense because people don't understand computers in the first place. – Pat Devlin Dec 14 '16 at 15:00
• It's not so much that the numbers should have some sort of "backstory" as that I think students should be able to compute them, or at least justify them based on some mathematical modelling, themselves. We saw an example of this where the die probabilities were derived based on symmetry (which arguably is a more elementary property than assigning specific values to each side). – tilo.wiklund Dec 14 '16 at 15:33
• [I thought the point was to move away from counting.] – Pat Devlin Dec 14 '16 at 19:34