Problem C. 944. (April 2008)

C. 944. Rick, Dan and Alan play table tennis with two players at one side of the table and the third one at the other side. Dan and Alan playing together beat Rick three times as often as Rick beats them, Dan wins the game against Rick and Alan as often as he loses it, and Alan wins the game against Rick and Dan twice as often as he loses it. Last time they played six games during the afternoon, two in each arrangement of players. What is the probability that Rick won at least once?

(5 pont)

Deadline expired on May 15, 2008.

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

Megoldás: A feltételekből kiderül, hogy Ricsi nyerési esélye Attila és Dénes ellen $\frac14$ ; Ricsi és Attila esélye Dénes ellen $\frac12$ ; végül Ricsi és Dénes esélye Attila ellen $\frac13$ .

Ezek alapján annak a valószínűsége, hogy Ricsi mind a hat meccsen veszített:

$\left(1-\frac14\right)\cdot\left(1-\frac14\right)\cdot\left(1-\frac12\right)\cdot\left(1-\frac12\right)\cdot\left(1-\frac13\right)\cdot\left(1-\frac13\right)=\frac34\cdot\frac34\cdot\frac12\cdot\frac12\cdot\frac23\cdot\frac23=\frac{1}{16}.$

Így annak valószínűsége, hogy legalább egyszer nyert: $p=1-\frac{1}{16}=\frac{15}{16}$ .

Statistics:

129 students sent a solution.

5 points: 63 students.

4 points: 11 students.

3 points: 7 students.

2 points: 5 students.

1 point: 24 students.

0 point: 19 students.

Problems in Mathematics of KöMaL, April 2008

129 students sent a solution.
5 points:	63 students.
4 points:	11 students.
3 points:	7 students.
2 points:	5 students.
1 point:	24 students.
0 point:	19 students.

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