Wikipedia:Reference desk/Archives/Mathematics/2016 February 22
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February 22
[edit]Robot baseball players...
[edit]If a baseball team were to consist of robot baseball players that struck out 70% of the time and walked 30% of the time, what is the average number of runs per inning they would get?Naraht (talk) 07:21, 22 February 2016 (UTC)
- Generalize a bit and say you get a walk with probability p and strike out with probability 1-p=q. The number of walks W you get before the third strikeout follows a negative binomial distribution; in this case
- 0: q3
- 1: 3pq3
- 2: 6p2q3
- 3: 10p3q3
- 4: 15p4q3
- etc.
- The number of runs is max(0, W-3) and we're looking for the expected value of this or
- 1⋅15p4q3 + 2⋅21p5q3 + 3⋅28p6q3 + ...
- which, according to my calculations, is
- p4q-1(15-18p+6p2).
- --RDBury (talk) 10:44, 22 February 2016 (UTC)
Zenithal map projection questions
[edit]Our article on the subject implies that it is possible to have a map projection of a sphere onto a plane that preserves direction from two points to every other point. However, I have found no link to a map projection that preserves directions for more than one point. Is our article wrong?--Leon (talk) 21:40, 22 February 2016 (UTC)
- I don't know if it has a name, but such a projection is easy to describe mathematically. Call your two special points p1 and p2. To know where a third point q on the sphere appears on the projection, measure the azimuth of q from p1 and from p2 on the sphere. On the projection, draw a line from p1 at the measured azimuth, and draw a line from p2 at the other measured azimuth. q lies at the intersection of the two lines. Egnau (talk) 16:03, 23 February 2016 (UTC)
- Just to clarify a bit, the article states this in the first bullet point in the section "Classification". Also note that this projection would only work for a patch, not the entire sphere. For example if p1 and p2 are on the equator and q is one of the poles then the directions would both be north/south and the third vertex of the triangle in the plane would be at infinity. In fact antipodal points would always map to the same point in the plane, so the projection would only work for a half-sphere at most. Also also note that p1 and p2 can't themselves be antipodal, otherwise the directions from q would be parallel again and the intersection in the plane would not be defined. Perhaps for these reasons, and the fact that such a projection would be highly distorted near the edges, this projection is rarely used in cartography and not notable in that sense. --RDBury (talk) 17:46, 23 February 2016 (UTC)
- Interesting. Thanks!
- Just to clarify a bit, the article states this in the first bullet point in the section "Classification". Also note that this projection would only work for a patch, not the entire sphere. For example if p1 and p2 are on the equator and q is one of the poles then the directions would both be north/south and the third vertex of the triangle in the plane would be at infinity. In fact antipodal points would always map to the same point in the plane, so the projection would only work for a half-sphere at most. Also also note that p1 and p2 can't themselves be antipodal, otherwise the directions from q would be parallel again and the intersection in the plane would not be defined. Perhaps for these reasons, and the fact that such a projection would be highly distorted near the edges, this projection is rarely used in cartography and not notable in that sense. --RDBury (talk) 17:46, 23 February 2016 (UTC)
- My next question (the title was "questions"): retroazimuthal projections allow someone to find the direction from any point B to a special point A. But, it's not clear how this would work with the Hammer retroazimuthal projection, for example, as the meridians are not straight lines. How would you go about taking an angle to define a bearing towards your chosen point? It's obvious with the Craig retroazimuthal projection as the meridians are vertical, but with the others I'm not so sure.--Leon (talk) 18:50, 23 February 2016 (UTC)
- You're supposed to align the "up" direction of the map with north, and get your bearing that way. The drawn meridians are only a distraction. In the Hammer example, if you take B = 15°S, 165°W and instead align the meridian at B with north, then the map would tell you to travel due north to reach A = 45°N, 90°W which is nonsense. Egnau (talk) 15:09, 24 February 2016 (UTC)
- Okay, thanks, I understand.
- You're supposed to align the "up" direction of the map with north, and get your bearing that way. The drawn meridians are only a distraction. In the Hammer example, if you take B = 15°S, 165°W and instead align the meridian at B with north, then the map would tell you to travel due north to reach A = 45°N, 90°W which is nonsense. Egnau (talk) 15:09, 24 February 2016 (UTC)
- My next question (the title was "questions"): retroazimuthal projections allow someone to find the direction from any point B to a special point A. But, it's not clear how this would work with the Hammer retroazimuthal projection, for example, as the meridians are not straight lines. How would you go about taking an angle to define a bearing towards your chosen point? It's obvious with the Craig retroazimuthal projection as the meridians are vertical, but with the others I'm not so sure.--Leon (talk) 18:50, 23 February 2016 (UTC)
- Also, I found a link to the two-point azimuthal projection. Should I create a page about it? The projection may not be 'obviously' notable, but given that the property of being two-point azimuthal is mentioned in the text of map projection it doesn't seem like such a bad idea. Does anyone know of any good software for producing maps of given projections?--Leon (talk) 21:27, 24 February 2016 (UTC)
- I don't think there's enough material there for a new page, and the site itself is apparently self-published so I doubt it would meet WP's criteria for reliability. It might help to add a link as a reference where the projection is mentioned in the article. Even if it's not a reliable source it's better than a cn flag. --RDBury (talk) 07:01, 27 February 2016 (UTC)
- Actually I found a more reliable source with more info and added it as a cite in the article. Finding this was a lot easier knowing the actual name of what to look for (go figure), so your link was a big help. Turns out this projection is rarely used and is equivalent to another better known projection, so still probably not notable enough for a new article. --RDBury (talk) 15:59, 27 February 2016 (UTC)
- I don't think there's enough material there for a new page, and the site itself is apparently self-published so I doubt it would meet WP's criteria for reliability. It might help to add a link as a reference where the projection is mentioned in the article. Even if it's not a reliable source it's better than a cn flag. --RDBury (talk) 07:01, 27 February 2016 (UTC)
- Also, I found a link to the two-point azimuthal projection. Should I create a page about it? The projection may not be 'obviously' notable, but given that the property of being two-point azimuthal is mentioned in the text of map projection it doesn't seem like such a bad idea. Does anyone know of any good software for producing maps of given projections?--Leon (talk) 21:27, 24 February 2016 (UTC)