Problem C. 1636. (November 2020)

C. 1636. The Hungarian poet Dezső Kosztolányi spent a few weeks in Paris when he was a student. When he was given for change a ten-centime coin not in circulation any more, he wanted to give it away. He did not succeed, which he explained to himself by the expression on his face revealing his intentions. Therefore he decided to get 9 valid ten-centime coins, mix them with the worthless coin in his pocket, and by not looking at them he pays with one of them in a shop. He continued doing so until he had a single coin in his pocket: the coin out of circulation. What is the probability of this?

(5 pont)

Deadline expired on December 10, 2020.

Sorry, the solution is available only in Hungarian. Google translation

1. megoldás. A 10 érmével \(\displaystyle 10!\) féle sorrendben fizethet, ez az összes lehetőségek száma. Azon sorrendek száma, amikor a forgalomból kivont marad meg utoljára \(\displaystyle 9!\), hiszen a 9 jó érmével ennyiféle sorrendben fizethetett, utoljára pedig a forgalomból kivont maradt. Tehát annak valószínűsége, hogy a forgalomból kivont marad meg utoljára \(\displaystyle 9!/10!=1/10\).

2. megoldás. Mivel a 10 érme (a forgalomból kivont és a 9 jó) bármelyike egyforma valószínűséggel marad meg utoljára, ezért ez a közös érték \(\displaystyle 1/10\).

Statistics:

67 students sent a solution.
5 points:	59 students.
4 points:	6 students.
1 point:	1 student.
Unfair, not evaluated:	1 solutions.

Problems in Mathematics of KöMaL, November 2020