52 BOOK OF MATHEMATICAL PEOBLEMS. 



234. Ten persons each write down one of the digits 0, 1, 2... 9 

 at random ; find the probability of all ten digits being written. 



235. A throws a pair of dice each of which is a cube; B 

 throws a pair one of which is a regular octahedron and the other a 

 regular tetrahedron whose faces are marked from 1 to 8, and from 

 1 to 4 respectively; which throw is likely to be the higher, the 

 number on the lowest face being taken in the case of the tetrahe- 

 dron 1 If A throws 6, what is the chance that B will throw 

 higher ? 



236. The sum of two positive quantities is known, prove that 

 it is an even chance that their product will be not less than three 

 fourths of their greatest possible product. 



237. Two points are taken at random on a given straight 

 line of length a: prove that the probability of their distance 



(f, \ 2 

 ) 



238. If three points be taken at random on the circum- 

 ference of a circle the probability of their lying on the same semi- 

 circle is -; . 



239. If q things be distributed among p persons, the chance 



that every one of the persons will have at least one is the coeffi- 



*_ 

 cicnt of of in the expansion of \q_(e p - l) p . 



240. If a rod be marked at random in n points and divided 

 at those points the chance that none of the parts shall be greater 



than - th of the rod is . 

 n n 



241. There are 2m black balls and m white balls from which 

 6 balls are drawn at random; prove that when m is very large 



