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

Question

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?

Answer 1

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 $\sum_{a \in A} P(\{a\}).$

Answer 2

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).

Answer 3

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]