Mathematical and Physical Journal
for High Schools
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Problem C. 1062. (January 2011)

C. 1062. A die is rolled n times. What is the probability that there are two equal numbers among the results obtained?

(5 pont)

Deadline expired on February 10, 2011.


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

Megoldás. Ha \(\displaystyle n\ge 7\), akkor a skatulye-elv miatt lesz két olyan dobás, mik egyenlőek, azaz \(\displaystyle P(n\ge 7)=1\).

Másrészről \(\displaystyle P(n\le 1)=0\).

Ha \(\displaystyle 2\le n\le 6\), akkor annak a valószínűsége, hogy minden dobás különböző \(\displaystyle \frac{6!}{(6-n)!6^n}\), azaz annak a valószínűsége, hogy lesz két egyforma \(\displaystyle P(2\le n\le 6)=1-\frac{6!}{(6-n)!6^n}\). Ez az egyes esetekben: \(\displaystyle P(n=2)=\frac 16\), \(\displaystyle P(n=3)=\frac 49\), \(\displaystyle P(n=4)=\frac{13}{18}\), \(\displaystyle P(n=5)=\frac{49}{54}\), \(\displaystyle P(n=6)=\frac{319}{324}\).


Statistics:

196 students sent a solution.
5 points:136 students.
4 points:3 students.
3 points:9 students.
2 points:27 students.
1 point:11 students.
0 point:4 students.
Unfair, not evaluated:6 solutions.

Problems in Mathematics of KöMaL, January 2011