Wikipedia:Reference desk/Archives/Mathematics/2012 December 15

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December 15

"≧" and "≦"

Both "≧" and "≦" redirect to Inequality (mathematics), and both symbols do indeed occur in the article, within the section "Vector inequalities". They are indeed explained there, perhaps. I say "perhaps" because my knowledge of maths is well below that assumed of the reader. Still, if I ignore a superscripted "n" after what I suppose is the set of real numbers, and superscripted "T"s after sets, I think I understand what's being said, which is that [crosses fingers] if x and y are vectors consisting of the same number of real numbers, then "x ≧ y" means "x_n ≥ y_n"; whereas "x ≥ y" means "x_n ≥ y_n, x ≠ y". Underinformed and hazy terminology aside, am I right so far?

The article then says "We note that this notation is consistent with that used by Matthias Ehrgott in Multicriteria Optimization (see References)". The denotation is clear enough, but as for the connotations: (i) "We note that" is "unencyclopedic" (or anyway unwikipedic) and even suggests that this sentence was lifted from elsewhere, and (ii) the fact that this is a 2005 book leaves the reader wondering whether the notation (a) is merely what is currently used or (b) was recently developed.

Anyway, if I can read out "x ≥ y" as "x  is greater than or equal to  y", how should I read out "x ≧ y"?

OR (backed up by [unsourced] ja-Wikipedia) tells me that in Japan, "≧" is (rightly or wrongly) the standard way to write "greater than or equal to". Is this specific to Japan, or is it also so used elsewhere?

(Before thinking of posting here, I asked about this here within the article's talk page; please reply either here or there, but perhaps not both.) -- Hoary (talk) 01:31, 15 December 2012 (UTC)

I replied on the article's talk page Talk:Inequality (mathematics)/Archive 1#"≧" and "≦". Let's take the discussion there. Duoduoduo (talk) 14:44, 15 December 2012 (UTC)

Thank you. Yes -- if anyone else is interested, do please comment there rather than here. -- Hoary (talk) 00:04, 16 December 2012 (UTC)

Cost:Benefit

If I have a small but calculable chance at a positive outcome, and a larger chance at a negative outcome, what is the process that determines when the potential benefit becomes better than the negative consequences? For example, if I have a 20% chance at gaining 10 dollars, but the bet is one dollar, how do I determine when the prize pot would no longer be worth the 80% chance at losing a dollar? Thanks. --66.188.84.18 (talk) 19:18, 15 December 2012 (UTC)

.2 × $10 =  $2.00
.8 × -$1 = -$0.80
            =====
            $1.20

So, under this setup, you gain $2 on average, and lose 80 cents, on average, so the net effect is to gain $1.20 per bet. That's a good bet. Also note that this assumes you get back the dollar you bet when you win, in addition to the $10 you win. If not, then the 80% chance of losing the dollar you bet goes up to 100%. You would still have a net gain of $1 per bet, though, so it's still a good bet. StuRat (talk) 19:31, 15 December 2012 (UTC)

Note, however, that probabilities alone are only part of the real-world answer. For example, if that $1 bet is needed to pay your mortgage and prevent foreclosure, you'd do best not to bet it. StuRat (talk) 19:37, 15 December 2012 (UTC)

I see, and so the bet becomes bad at a $3 prize and breaks even at $4. Thanks a lot for your help. --66.188.84.18 (talk) 19:39, 15 December 2012 (UTC)

You're welcome. StuRat (talk) 19:46, 15 December 2012 (UTC)

What StuRat described is called the expectation of the money you receive. However, this only works for small amounts. For larger amounts you need to consider the expectation of the utility instead. -- Meni Rosenfeld (talk) 06:36, 16 December 2012 (UTC)