document.write( "Question 1208705: The Boomtown Bears are playing against the Tipton Toros in a baseball tournament. The winner of the tournament is the first team that wins three games. The Bears have a probability of $\frac{1}{2}$ of winning each game. Find the probability that the Bears win the tournament. Assume there are no ties—every game has a winner.
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Algebra.Com's Answer #847110 by ikleyn(52786) $\"\"$ $\"About$

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\n" ); document.write( "The Boomtown Bears are playing against the Tipton Toros in a baseball tournament.
\n" ); document.write( "The winner of the tournament is the first team that wins three games.
\n" ); document.write( "The Bears have a probability of 1/2 of winning each game.
\n" ); document.write( "Find the probability that the Bears win the tournament.
\n" ); document.write( "Assume there are no ties—every game has a winner.
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document.write( "We have two teams: B (Bears) and T (the other team).\r\n" );
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document.write( "First, from the condition, it is clear that every team has the probability 1/2 to win/(to lose) \r\n" );
document.write( "every individual game.\r\n" );
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document.write( "Next, there are not too many cases when the Bear wins the tournament.\r\n" );
document.write( "It is easy to list and to analyze each of these cases individually.\r\n" );
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document.write( "Case 1. The Bear wins the tournament after the 3 first games.\r\n" );
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document.write( "        It means that the Bear wins 3 of the 3 first games; the other team loses all three games.\r\n" );
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document.write( "        The winning record of the tournament for Bears is BBB : the Bears wins all three games of the first 3 games.\r\n" );
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document.write( "        The probability of such outcome is P(case 1) =   = .    (1)\r\n" );
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document.write( "Case 2. The Bears wins after the 4 first games (but does not win after 3 first games).\r\n" );
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document.write( "        It means that the Bears wins 3 of 4 games; the other team wins 1 game.\r\n" );
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document.write( "        The winning records of the tournament for the Bears are BBTB,  BTBB,  TBBB.\r\n" );
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document.write( "               Notice that in case 2 \"T\" can not be in the last position (!)\r\n" );
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document.write( "        \r\n" );
document.write( "        The probability for the Bears to win in this case is\r\n" );
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document.write( "            P(case 2) =  = .    (2)\r\n" );
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document.write( "Case 3. The Bear wins after 5 games (but does not win after 3 or 4 games).\r\n" );
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document.write( "        It means that the Bears wins 3 of 5 games; the other team wins 2 games.\r\n" );
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document.write( "        The winning records of the tournament for the Bears have 5 positions; the last position must be B and can not be T.\r\n" );
document.write( "        \r\n" );
document.write( "               So, two \"T\" can occupy any of 4 remaining positions, and it provides   =  = 6 possible winning records for B.\r\n" );
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document.write( "        Therefore, the probability for the Bears to win in this case is\r\n" );
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document.write( "            P(case 3) =   =  = .    (3)\r\n" );
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document.write( "It is easy to understand that these three cases exhaust all possibilities \r\n" );
document.write( "when the Bears wins the tournament.\r\n" );
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document.write( "Our last step is to add the three found probability (1), (2) and (3) for the Bears to win the tournament\r\n" );
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document.write( "    P = P(case 1) + P(case 2) + P(case 3) =  +  +  =  =  = .\r\n" );
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document.write( "ANSWER.  The probability for the Bears to win this tournament is   = 0.5.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Probably, this answer could be predicted without calculations.
\n" ); document.write( "Indeed, some of the two teams must be the first winning three games, and there is a symmetry between the teams.
\n" ); document.write( "So, the probability is 1/2 for each team to win the tournament.\r
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