The full post is at:\u00a0 http:\/\/fivethirtyeight.com\/features\/which-geyser-gushes-first\/<\/a><\/p>\n and a complete solution is in the next week’s post:<\/p>\n http:\/\/fivethirtyeight.com\/features\/how-long-will-your-smartphone-distract-you-from-family-dinner\/<\/a><\/p>\n But what I want to discuss is a different way of thinking about the solution than the one posted at 538.\u00a0 I want to use geometric probability.\u00a0 [If you want to solve this on your own, spoilers ahead.]<\/p>\n Let’s start with an easier version of the question…<\/strong><\/em><\/p>\n Suppose the only geysers were A and B.<\/p>\n If you have just arrived, Geyser A will erupt in x hours, where x is between 0 and 2. And Geyser B will erupt in y hours, where y is between 0 and 4.\u00a0 Intuitively, it seems likely that y will be greater than x.\u00a0 One way to see this is to use geometric probability.\u00a0 Think of every possible outcome as an ordered pair.\u00a0 Then, all possible outcomes can be represented as an area on the coordinate plane.\u00a0 I used Desmos.com to graph the region and also the line y=x, or more precisely, the shaded half-plane y>x.<\/p>\n Geyser A will erupt first whenever y>x.\u00a0So the probability\u00a0we seek is the area of the part of the box above the x=y line divided by the total area of the box. \u00a0You can just about count boxes to determine that this occurs 6\/8 (or 3\/4)\u00a0of the time.<\/p>\n OK then.\u00a0 Now let’s bring Geyser C back into the problem…<\/strong><\/em><\/p>\n From the moment you arrive, you know that Geyser C will erupt in z hours, where 0<z<6.\u00a0 But you can still use geometric probability.\u00a0 Each outcome can be thought of as an ordered triple (x,y,z).\u00a0 The space of all outcomes is now a 2 x 4 x 6 rectangular solid.\u00a0 And if you are looking for the probability\u00a0that Geyser A erupts first, you want to slice that solid up with the planes x=y and x=z.<\/p>\n I don’t know if Desmos can do this kind of thing.\u00a0 But fortunately, Paul Seeburger’s Multivariable Calculus Exploration applet\u00a0can be used to visualize this problem.\u00a0 You can find the applet here:<\/p>\n http:\/\/web.monroecc.edu\/manila\/webfiles\/calcNSF\/JavaCode\/CalcPlot3D.htm<\/a><\/p>\n Note: on my computer, I can only open this page in Firefox.\u00a0 YMMV.<\/p>\n Here is the solid that represents the set of all possible outcomes: And here is that same solid, cut by the planes x=y and x=z: We need to find the volume of the part of this box “to the right” of the\u00a0x=y plane and “above” the x=z plane. It’s easier to see what we are looking for if you rotate the diagram to\u00a0view it “from behind”:<\/p>\n In this diagram, the volume defined by points where y>x and z>x can be broken into\u00a0four chunks.\u00a0 I’ve added the x=2 plane and the z=2 plane to help you see them.<\/p>\n The rectangular solid on the upper left: V = 2x2x4=16<\/p>\n The triangular prism in the front on the lower left: V = (1\/2)x2x2x2=4<\/p>\n The triangular prism in the front on the upper right: V = (1\/2)x2x2x4 = 8<\/p>\n The\u00a0front half of the pyramid on the bottom right: V = (1\/3)(2)(2)(2) = 8\/3<\/p>\n (If you are having trouble seeing the pyramid: in this view, we are looking through its 2×2 base, looking toward its vertex which is 2 units away.)<\/p>\n The probability of Geyser A erupting first is the sum of these volumes divided by the total volume of the box, or:<\/p>\n (16 + 4+ 8 + 8\/3)\/(2x4x6) which reduces to 23\/36.<\/p>\n Using a similar method but with different cutting planes, you can find the pictures that will give you the probabilities of Geyser B or Geyser C erupting first.\u00a0 And you do get the same answer that is posted on the Riddler’s webpage.<\/strong><\/em><\/p>\n You can post questions (or ask\u00a0me in class) if you\u00a0need help with those last two…<\/p>\n <\/p>\n <\/p>\n","protected":false},"excerpt":{"rendered":" Hello all…sorry the posts are coming at such long intervals.\u00a0 I am teaching a new class this year — multivariable calculus! — and it is taking up lots of time as I try to remember\/recreate\/invent some level of understanding, having … Continue reading ![\"2d](\"http:\/\/advancedmathyoungstudents.com\/blog\/wp-content\/uploads\/2015\/12\/2d-geyser.png\")
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