When you roll a die, do you expect it to turn up three-and-a-half?
Hell no. But that still is the expected value of a die roll, I’m afraid.
What does that mean?
Here’s what it means. If you roll that die often enough, you’ll get 1 one sixth of the time, and 2 one sixth of the time, etc. Add those up and you get (1+2+3+4+5+6) / 6 = 3.5 and that’s what mathematics calls the expected value, or expectation (Icelandic: “væntigildið,” or “vænta gildið”) of something random.
That makes no sense, until the mathematician explains that he really means the expected average value. If you keep rolling that die until the sound of it drives you bonkers, writing down each result you get, then you should expect your average to be pretty close to three-and-a-half.
So on any one roll of a die, whatever you expect, you shouldn’t expect to get the expected value. That would be pretty unexpected.
[Incidentally, when you roll a pair of dice, you should expect to get the expected value, which is 7. That's because for a pair of dice, the expected value is both (a) actually possible, and (b) the most probable one. 6 and 8 are a little less likely, 5 and 9 still less likely, and so forth. 2 and 12 are really unlikely. If you get to gamble on a pair of dice with even odds, bet on 7 every time. Trust me on this.]
My point? Ok, this posting is not really about mathematics. It is about language.
The term “expected value” means a slightly different thing in mathematical English than it might mean in other dialects of English.
More generally, a mathematical background changes a person’s use of language ever-so-slightly.
Still more generally, language differs with background. Culture. Upbringing. Education. Individual temperament. The way you woke up that morning. Language differs with a lot of factors. In some sense, no two people speak quite the same language.
If you do not keep this in mind when talking to someone, you will misunderstand them. Even if you do keep it in mind, you will still misunderstand them a bit. It’s a basic human function; we breathe, eat, urinate, copulate, and misunderstand.
We do have this jolly nice invention, language, for transferring ideas and concepts and feelings from one mind to another by way of speech (and writing), and that’s pretty cool, but it’s far from perfect. We do a fair job of it for simple things (“pass me the raita please”), but for anything more complex or more important (“I like you”), the message often does not get across in quite the way it was intended.
In a restaurant today, my father called for the attention of the waitress with a chipper “Fröken!” (Miss!) and I gently explained that nowadays a lot of Icelanders in service jobs find that appellation offensive. How was he supposed to know? It was the polite way to call for attention when he was growing up. Even my brother-in-law, hardly even older than me, didn’t know. But I’ve heard several people in service jobs make it pretty clear that in their language, it is plain rude.
Ask two people to define exactly what they mean by a word, and their definitions will differ. Find several people coming out of a meeting and ask them what went on, and their responses will vary, sometimes wildly, even though they all heard the exact same words spoken in there.
When someone’s attitude or opinions or reactions completely fail to meet your, hm, expectations, you may find it useful to consider the difference between their frame of reference and yours.
You might not be speaking quite the same language.