Difference between revisions of "1996 AIME Problems/Problem 6"
m (→Solution) |
m (→Solution) |
||
Line 13: | Line 13: | ||
Similarity, the probability that one team loses all games is <math>\frac{5}{16}</math>. | Similarity, the probability that one team loses all games is <math>\frac{5}{16}</math>. | ||
− | The probability that one team wins all games and another team loses all games is <math>(5\cdot (\frac{1}{2})^4)(4\cdot (\frac{1}{2})^3)=\frac{5}{32}</math>. | + | The probability that one team wins all games and another team loses all games is <math>\left(5\cdot \left(\frac{1}{2}\right)^4\right)\left(4\cdot \left(\frac{1}{2}\right)^3\right)=\frac{5}{32}</math>. |
<math>\frac{5}{16}+\frac{5}{16}-\frac{5}{32}=\frac{15}{32}</math> | <math>\frac{5}{16}+\frac{5}{16}-\frac{5}{32}=\frac{15}{32}</math> |
Revision as of 21:31, 20 December 2018
Problem
In a five-team tournament, each team plays one game with every other team. Each team has a chance of winning any game it plays. (There are no ties.) Let
be the probability that the tournament will produce neither an undefeated team nor a winless team, where
and
are relatively prime integers. Find
.
Solution
We can use complementary counting: finding the probability that at least one team wins all games or at least one team loses all games.
No more than 1 team can win or lose all games, so at most one team can win all games and at most one team can lose all games.
Now we use PIE:
The probability that one team wins all games is .
Similarity, the probability that one team loses all games is .
The probability that one team wins all games and another team loses all games is .
Since this is the opposite of the probability we want, we subtract that from 1 to get .
See also
1996 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.