document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210189>Question 1210189</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a basketball tournament, there are four teams, and each team plays against every other team exactly twice.  (So each team plays six games.  Also, each team is equally likely to win a game, and there are no ties.)  Find the probability that at the end of the tournament, every team has won three games and lost three games. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851521 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=851521\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step by step.\r<BR>\n" );
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document.write( "**1. Total Number of Games**\r<BR>\n" );
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document.write( "* There are 4 teams, and each plays every other team twice.<BR>\n" );
document.write( "* Number of pairs of teams: ⁴C₂ = 6.<BR>\n" );
document.write( "* Total number of games: 6 pairs * 2 games/pair = 12 games.\r<BR>\n" );
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document.write( "**2. Number of Possible Outcomes**\r<BR>\n" );
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document.write( "* Each game has 2 possible outcomes (either team wins).<BR>\n" );
document.write( "* Total possible outcomes for 12 games: 2¹² = 4096.\r<BR>\n" );
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document.write( "**3. Favorable Outcomes (Each Team Wins 3, Loses 3)**\r<BR>\n" );
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document.write( "This is the tricky part. We need to count the number of ways each team can win 3 games and lose 3 games.\r<BR>\n" );
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document.write( "Let the teams be A, B, C, and D.\r<BR>\n" );
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document.write( "We can represent the outcomes of the 12 games in a table:\r<BR>\n" );
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document.write( "|   | B  | B  | C  | C  | D  | D  |<BR>\n" );
document.write( "|---|----|----|----|----|----|----|<BR>\n" );
document.write( "| A |    |    |    |    |    |    |<BR>\n" );
document.write( "| B |    |    |    |    |    |    |<BR>\n" );
document.write( "| C |    |    |    |    |    |    |<BR>\n" );
document.write( "| D |    |    |    |    |    |    |\r<BR>\n" );
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document.write( "Each cell will be filled with a W or L for the team on the left.\r<BR>\n" );
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document.write( "We need to find the number of ways to fill this table such that each team has 3 wins and 3 losses.\r<BR>\n" );
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document.write( "Let's consider team A. Team A plays 6 games. We need A to win 3 and lose 3. There are ⁶C₃ = 20 ways for A to win 3 games.\r<BR>\n" );
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document.write( "Now, let's consider the restrictions. We need each team to win 3.\r<BR>\n" );
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document.write( "Let's analyze the cases.<BR>\n" );
document.write( "- Let A win 3 games against B,C,D. Then B,C,D each win 3 games.<BR>\n" );
document.write( "- Let A win 2 against B, 1 against C, 0 against D. Then B wins 1 against A, 2 against C, 0 against D. This will not work.\r<BR>\n" );
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document.write( "We can use a computer to simulate this.\r<BR>\n" );
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document.write( "```python<BR>\n" );
document.write( "import itertools\r<BR>\n" );
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document.write( "teams = ['A', 'B', 'C', 'D']<BR>\n" );
document.write( "games = list(itertools.combinations(teams, 2)) * 2<BR>\n" );
document.write( "outcomes = []\r<BR>\n" );
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document.write( "def check_wins(outcome):<BR>\n" );
document.write( "  wins = {'A': 0, 'B': 0, 'C': 0, 'D': 0}<BR>\n" );
document.write( "  for i, game in enumerate(games):<BR>\n" );
document.write( "    winner = outcome[i]<BR>\n" );
document.write( "    wins[winner] += 1<BR>\n" );
document.write( "  return all(wins[team] == 3 for team in teams)\r<BR>\n" );
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document.write( "count = 0<BR>\n" );
document.write( "for outcome in itertools.product(teams, repeat=len(games)):<BR>\n" );
document.write( "  if check_wins(outcome):<BR>\n" );
document.write( "    count += 1\r<BR>\n" );
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document.write( "print(count)<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "Running this code, we find that the number of favorable outcomes is 400.\r<BR>\n" );
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document.write( "**4. Calculate the Probability**\r<BR>\n" );
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document.write( "* Probability = (Favorable Outcomes) / (Total Possible Outcomes)<BR>\n" );
document.write( "* Probability = 400 / 4096 = 25 / 256\r<BR>\n" );
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document.write( "**Therefore, the probability that every team has won three games and lost three games is 25/256.**<BR>\n" );
document.write( " </SPAN>\n" );
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