Question

in a tournament 5 teams play each other once with a win or loss prob of winning is 1/2 probabiltiy that no team wins all matches or loses all

• A. 1/2
• B. 17/32
• C. 1/4
• D. 3/4

Answer 1

Answer: (B)

17/32

Answer 2

Here's how to solve this probability problem:

1. Total Possible Outcomes:

   • There are a total of 10 matches played in the tournament, calculated as $\binom{5}{2} = 10$.
   • Each match has two possible outcomes (win or loss), so the total number of outcomes is $2^{10} = 1024$.

2. Extreme Cases (Team Wins All or Loses All):

   • For a single team to win all 4 of its matches, there is only one way for those matches to turn out. The remaining 6 matches can have any outcome, resulting in $2^6 = 64$ possibilities.
   • Similarly, for a single team to lose all 4 of its matches, there are also $2^6 = 64$ possibilities.
   • Naively summing these possibilities for all 5 teams gives $5 \times (64 + 64) = 640$.

3. Inclusion-Exclusion Principle (Correcting for Overcounting):

   • We've overcounted the cases where one team wins all its matches and another team loses all its matches.
   • For a fixed pair of teams (one winning all, one losing all), the matches involving these two teams are fixed. There are $\binom{3}{2} = 3$ matches remaining among the other 3 teams, which can have $2^3 = 8$ outcomes.
   • There are 5 choices for the winning team and 4 choices for the losing team, resulting in $5 \times 4 \times 8 = 160$ such cases.
   • Using the inclusion-exclusion principle, the number of outcomes with at least one extreme record is $640 - 160 = 480$.

4. Favorable Outcomes (No Team Wins All or Loses All):

   • The number of outcomes where no team wins all its matches or loses all its matches is the total number of outcomes minus the number of outcomes with at least one extreme record: $1024 - 480 = 544$.

5. Probability Calculation:

   • The probability that no team wins all its matches or loses all its matches is the number of favorable outcomes divided by the total number of outcomes: $\frac{544}{1024} = \frac{17}{32}$.

Therefore, the probability that no team wins all matches or loses all matches is $\frac{17}{32}$.