Wikipedia:Reference desk/Archives/Mathematics/2011 November 17

= November 17 =

Probability Question
Hi I'm preparing for a stats test and I'm stumped on a problem I found in my text (paraphrased below):

2. There are n children in a classroom and each child has exactly one toy (so there are n toys in total). They break for recess, leaving their toys in the classroom. When they come back to the classroom, each child picks a toy by random. a) What the expected number of children who pick the toy they had before leaving for recess? b) What's the probability no child picks the toy they had initially? — Preceding unsigned comment added by Rain titan (talk • contribs) 10:14, 17 November 2011 (UTC)
 * See Derangement. Bo Jacoby (talk) 10:39, 17 November 2011 (UTC).


 * The wording of the question is not very clear: what does it mean for the children to pick a toy at random? Do they each independently name the one they want (so that a toy could be chosen by more than one child), or do they choose randomly in sequence from the toys that are left after those before them in the queue have chosen? (The reality is of course likely to be more disorganised and tearful than either of these.) AndrewWTaylor (talk) 12:22, 17 November 2011 (UTC)
 * Agreed, bring cookies to help distract the children.Naraht (talk) 16:53, 17 November 2011 (UTC)
 * True. However, this wording or something rather similar is standard in text books introducing derangements. JoergenB (talk) 19:30, 17 November 2011 (UTC)
 * Sorry for the late reply. Each child chooses a top left after those before them in the queue have chosen (so each child will have exactly one toy). Thanks! — Preceding unsigned comment added by Rain titan (talk • contribs) 04:38, 18 November 2011 (UTC)
 * See Derangement. --COVIZAPIBETEFOKY (talk) 14:14, 18 November 2011 (UTC)

Inscribed circle question
Take a semi-circle whose flat side has a length of 4 units. The largest circle which can be inscribed in the semicircle touches the middle of the arc and the middle of the flat side and has radius 1 (diameter 2). What is the radius of the inscribed circle in each of the two shark fin shaped pieces between the Semi-circle and the inscribed diameter 1 circle?Naraht (talk) 16:51, 17 November 2011 (UTC)
 * See Descartes' theorem, especially the section on special cases. --RDBury (talk) 21:11, 17 November 2011 (UTC)
 * Ignore that, it doesn't apply in this case.--RDBury (talk) 21:21, 17 November 2011 (UTC)
 * This is an example of the Problem of Apollonius. There may be a theorem that covers this case nicely but if there is I can't find it at the moment. In any case, the problem can be solved with Inversive geometry. Apply the inversion with respect to the semicircle, it and the diameter remain fixed and the smaller circle becomes a line parallel to the diameter and tangent to the semicircle. You must find a circle tangent to the lines, so radius 1, and tangent to the semicircle, so the distance from the center to the center of the semicircle is 3. Inverting again to get the solution to the original problem, the desired circle has a diameter whose endpoints are at distance 1 and 2 from the center of the semicircle and when extended it passes through that point. So the radius of the desired circle is 1/2.--RDBury (talk) 21:57, 17 November 2011 (UTC)


 * Put the drawing into Cartesian coordinates, so the diameter is in the X axis and its middle point is (0,0). Let x, y, r be the coordinates of the centre and the radius of a circle S wee seek. S is tangent to two given circles and the X axis, so x, y, r must satisfy $$

\left\{ \begin{array}{ccc} x^2+y^2 & = & (2 - r)^2 \\ x^2+(y-1)^2 & = & (1 + r)^2 \\ y & = & r \end{array} \right. $$ This results (if I did my calculations right) in solutions: $$(x, y, z) = (\pm\sqrt 2, 1/2, 1/2)$$. Replacing the third equation with $$|y| = r$$, which is more general expression for 'circle being tangent to the X axis' results in thhe third solution $$(0, -1, 1)$$, which is, of course, outside the semicircle. CiaPan (talk) 06:59, 21 November 2011 (UTC)