Wikipedia:Reference desk/Archives/Mathematics/2013 November 17

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November 17

What is ⦞?

Can someone explain what the symbol ⦞ is used for? It's not in the list of mathematical symbols! - Letsbefiends (talk) 03:04, 17 November 2013 (UTC)

Making this very much larger: ⦞ - Letsbefiends (talk) 03:04, 17 November 2013 (UTC)

This symbol is part of the Unicode Miscellaneous Mathematical Symbols-B which has a number of unusual symbols. The article Mathematical operators and symbols in Unicode has a nice collection of symbols. I've personally never seen or used this symbol. Some of the Unicode codepoints are...whimsical at best. The Angle#Intersecting angle pairs section seems to claim that the symbol stands for "angle+s" = "angles" --Mark viking (talk) 04:25, 17 November 2013 (UTC)

I saw this in a Dover publication on trigonometry, but I couldn't locate what it meant. I don't think it's whimsical, it has some sort of obscure meaning. - Letsbefiends (talk) 05:36, 17 November 2013 (UTC)

My goodness, you are correct! That is exactly what it means - a plurality of angles! - Letsbefiends (talk) 05:47, 17 November 2013 (UTC)

L. E. Dickson in History of the Theory of Numbers (esp. Vol. 2) used the square version of this to refer to the square of an unspecified integer. Also, it sometimes is used to mean Q.E.D. Duoduoduo (talk) 17:29, 17 November 2013 (UTC)

Wrong. The rectangle you see is a result of missing glyph for Unicoded character.
The character actually looks like --CiaPan (talk) 10:53, 18 November 2013 (UTC)

Found three fonts that display the glyph correctly: Cambria, Cambria Math and Segoe UI Symbol. --CiaPan (talk) 12:17, 20 November 2013 (UTC)

math probablity

if MN no in a matrix are independently and randomly selected form a continues distribution what is probability of dating a saddle point (there is a no which is both minimum of it,s row and maximum of is,s columes — Preceding unsigned comment added by 117.241.137.28 (talk) 09:14, 17 November 2013 (UTC)

There seems to be a lot of typos in the question so let me try a paraphrase: Given an m by n matrix whose entries are chosen randomly and independently from a continuous probability distribution, what is the probability of there being a saddle point? Here a saddle point is defined as an entry which is both the minimum of its row and the maximum of its column. Say so if this is not what you meant.

First compute the probability that a particular entry say a₁₁ is a saddle point. Since the distribution is continuous we can safely ignore the possibility that two entries are the same. Also, the entries not in row 1 or column 1 don't affect the outcome so we can ignore them and look at just the m+n-1 entries a₁₁, a12, ... a_1n, a₂₁, ... a_m1. Finally, we only care about the order of the entries and not the actual values, so we can take the entries to be some random permutation of {1, 2, ..., m+n-1}. So the question boils down to: Given a random permutation [s₁, s₂, ... , s_m+n-1] of {1, 2, ..., m+n-1}, what is the probability that each s_i > s₁ for i in {2, ..., n} and s_i < s₁ for i in {n+1, ..., m+n-1}. This happens if and only if s₁=n, {s₂, ..., s_n}={1, ..., n-1} and {s_n+1, ..., s_m+n-1}={n+1, ..., m+n-1}. The number of ways this can happen is (n-1)!(m-1)! so the probability is (n-1)!(m-1)!/(m+n-1)!. (The connection with the beta function can be seen by looking at the uniform distribution on [0, 1].) The probability of a given entry being a saddle point being found, the question now at hand is to find the probability that any entry is a saddle point. It's easy to show that there cannot be 2 saddle points in a matrix, so this is mn times the probability that a given entry is a saddle point. That is n!m!/(m+n-1)!. Note that for m or n = 1 this works out to 1 as expected. --RDBury (talk) 15:15, 17 November 2013 (UTC)

Follow-up to "Question about probability"

This question is a follow-up to my previous question on this Reference Desk (located at this link: Wikipedia:Reference desk/Archives/Mathematics/2013 November 11#Question about probability). In the original question (at that link), I had asserted that: "When a person goes to sleep with his contact lenses in, he increases his chances of getting an eye infection by 10%". I had asked if this means that after ten nights of sleeping with contact lenses in, the probability of developing an eye infection would be 100% (since 10 nights multiplied by 10% each = 100%). The responses and the rest of the discussion follow at that link above. However, I would like to pose that same question, as above, but to modify the original premise. So, I am here simply changing the semantics involved in the original question. Would the analysis and probability results be different if the original premise had stated: "When a person goes to sleep with his contact lenses in, his chances of getting an eye infection are 10%" ... ? So, I am changing the premise that the probability increases by ten percent (of the original probability) to simply the probability is now exactly a flat-out ten percent chance. Thanks. Joseph A. Spadaro (talk) 23:36, 17 November 2013 (UTC)

If we assume that that means 10% per night, and that each night's probability is independent of the previous nights results (neither of which is likely a valid assumption, BTW), then the formula is:

P = 100×(1.0-(0.9)D)

So, after 10 days, the probability of having an eye infection would be:

P = 100×(1.0-(0.9)10) = 65.13%

To help understand the math, let's just look at two nights. Here the above formula gives us a 19% chance of having an eye infection. This breaks down as follows:

EYE INFECTION BOTH NIGHTS       = 10% × 10% = 1% 
EYE INFECTION ON 1ST NIGHT ONLY = 10% × 90% = 9%
EYE INFECTION ON 2nd NIGHT ONLY = 90% × 10% = 9%
NO EYE INFECTION                = 90% × 90% = 81%

Note that the top three lines total a 19% chance of an eye infection on one or both nights. StuRat (talk) 00:44, 18 November 2013 (UTC)

Thanks. OK, that all makes a lot of sense. I see what you did. Thanks. Now, to add a wrinkle. In the above list, your second item says "Eye Infection on the First Day Only" (and, thus, not on the second day). But, if you have an eye infection on the first day, (by definition) you will also have it on the second day (i.e., not a "new" eye infection, but the same one from the previous day). That is, let's assume that an eye infection does not clear/resolve simply in 24 hours. Does this "fact" change all the math that you did above? Thanks. Joseph A. Spadaro (talk) 01:52, 18 November 2013 (UTC)

This is where we get to my earlier point that having an eye infection on one particular night is not independent of having it on another. With my above calculation, the top two lines would then be combined into "EYE INFECTION ON BOTH NIGHTS", at a 10% chance. This does not affect the chance of not having an eye infection at all, however, or of ONLY having it on the last night. If there were more than 2 nights, then the chances of having an eye infection on the middle nights would also go up, but the chances of having one on the last night ONLY or not at all would still be the same. Of course, the chances of having an eye infection on the last night, in total, go up, as this includes having it left over from an earlier occurrence. StuRat (talk) 20:47, 18 November 2013 (UTC)

OK, thanks. Yes, that all makes sense. Thanks for the detailed explanation. It was very helpful. Thanks! Joseph A. Spadaro (talk) 13:50, 19 November 2013 (UTC)