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I'm trying to understand joint probabilities and dependence/independence. I've used examples with coins and cards, but when it comes to real life, I mix them up again.

I want to think of real life examples that are:

  1. Disjoint/independent
  2. Disjoint/dependent
  3. Joint/independent
  4. Joint/dependent

Here are my examples, but for the first one I'm stuck.

  1. ?
  2. Night and day
  3. Voting for Trump and liking milk
  4. Voting for Trump and being pro-gun

Are these correct? And what are some real life examples of independent and disjoint events?

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2 Answers 2

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Disjoint/independent: Disjoint events are only statistically independent if each outcome has probability zero or one (which is a trivial case). That is the reason you are having trouble finding an example of this case; it is a pathological case that is not important.

Non-disjoint/independent: You need to be careful with this one, since causal independence does not imply statistical independence. As a rough rule of thumb (which is slightly exaggerated, but not much) everything is correlated with everything. Although they are unlikely to be causally related, it would surprise me if voting-preference is uncorrelated with milk-consumption. It is more likely that rates of milk-consumption differ for Trump-voters and non-Trump voters, for all sorts of indirect reasons. For example, rates of milk-consumption among US adults differ by age and sex, and voting patterns also differ by age and sex. In particular, within the population of US adults, males drink more milk, and are also more likely to vote for Trump, so if this is one of the main indirect effects, I would expect that there would be some small positive correlation between milk-consumption and voting for Trump.

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  • $\begingroup$ Technically isn't there also the trivial case were they both have probability zero? They are still disjoint technically because their intersection is still empty. $\endgroup$
    – Joseph Garvin
    Commented Jun 18, 2018 at 15:45
  • $\begingroup$ That is true in general, but I was assuming he is talking about binary events, where they partition the sample space. In the latter case one event having probability zero means that the other has probability one. You are right though: in the more general case beyond a binary partition, you could have multiple disjoint events that don't cover the sample space and each have probability zero. I have edited to cover the more general case. $\endgroup$
    – Ben
    Commented Jun 19, 2018 at 0:10
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You're stuck because being both disjoint and independent is impossible (except for the corner case Ben points out where one of the events never happens under any circumstances). Disjoint means the two events are mutually exclusive -- if one happens than the other can't happen. Independent means if one happens it doesn't affect whether or not the other happens. You can't simultaneously prevent the other from happening and also not affect whether it happens. Disjoint implies dependent.

Your other examples look correct (at least in spirit, I have not surveyed voter milk drinking habits).

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