Mathematical and Physical Journal
for High Schools
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Problem A. 381. (October 2005)

A. 381. A die of n faces is rolled until each of the n possible outcomes is obtained at least once. What is the expected value of the number of rolls needed?

(5 pont)

Deadline expired on November 15, 2005.


Sketch of solution. It is well-known that the expected number of required experiments to reproduce an event of probability p is exactly 1/p.

If we have k different results of the possible n, we expect another n/(n-k) experiments to produce a new result.

The excpected number of all experiments together is n\left(\frac11+\frac12+\dots+\frac1n\right).


Statistics:

26 students sent a solution.
5 points:Balambér Dávid, Csóka Győző, Dücső Márton, Erdélyi Márton, Fischer Richárd, Gyenizse Gergő, Hujter Bálint, Jankó Zsuzsanna, Kisfaludi-Bak Sándor, Kónya 495 Gábor, Korándi Dániel, Nagy 224 Csaba, Nagy 317 Péter, Nagy Gergely Gábor, Paulin Roland, Radnai András, Szűcs Gergely, Tomon István, Udvari Balázs.
4 points:Ureczky Bálint.
3 points:1 student.
2 points:1 student.
0 point:4 students.

Problems in Mathematics of KöMaL, October 2005