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Difference between revisions of "1996 AIME Problems/Problem 6"

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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>.
  
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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>.
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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 20:31, 20 December 2018

Problem

In a five-team tournament, each team plays one game with every other team. Each team has a $50\%$ chance of winning any game it plays. (There are no ties.) Let $\dfrac{m}{n}$ be the probability that the tournament will produce neither an undefeated team nor a winless team, where $m$ and $n$ are relatively prime integers. Find $m+n$.

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 $5\cdot \left(\frac{1}{2}\right)^4=\frac{5}{16}$.

Similarity, the probability that one team loses all games is $\frac{5}{16}$.

The probability that one team wins all games and another team loses all games is $\left(5\cdot \left(\frac{1}{2}\right)^4\right)\left(4\cdot \left(\frac{1}{2}\right)^3\right)=\frac{5}{32}$.

$\frac{5}{16}+\frac{5}{16}-\frac{5}{32}=\frac{15}{32}$

Since this is the opposite of the probability we want, we subtract that from 1 to get $\frac{17}{32}$.

$17+32=\boxed{049}$

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

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