document.write( "Question 1119657: Two containers A and B are different. The Container A contains three white balls and two red balls and container B contains only one white ball. A coin is thrown and if faced, a ball is randomly withdrawn from container A and placed in container B, but if it yields a crown, then two balls are randomly withdrawn from container A and placed in container B. Now a ball is withdrawn randomly from container B. Question: Which the probability of the ball to be white? \n" ); document.write( "

Algebra.Com's Answer #735276 by ikleyn(52780) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Let's analyse first what will happen, when the coin is faced up.\r
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document.write( "1.  With the probability of 1/2 the coin is faced up.\r\n" );
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document.write( "        Then  with the probability 3/5 White ball goes from A to B, making B-content equal to (2W,0R),\r\n" );
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document.write( "          OR  with the probability 2/5  Red ball  goes from A to B, making B-content equal to (1W,1R).\r\n" );
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document.write( "    Summary for this step:\r\n" );
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document.write( "            With the probability (1/2)*(3/5) = 3/10 the content of B becomes (2W,0R)    (1a)    and/or\r\n" );
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document.write( "            with the probability (1/2)*(2/5) = 2/10 the content of B becomes (1W,1R).   (1b)\r\n" );
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document.write( "     Next step is to withdraw the ball from B. \r\n" );
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document.write( "          In case (1a) we withdraw White ball with the probability 1.\r\n" );
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document.write( "          In case (1b) we withdraw White ball with the probability 1/2.\r\n" );
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document.write( "     Thus the final probability to have White ball as the final outcome at this consideration is\r\n" );
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document.write( "          +  =  = .  (*)\r\n" );
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document.write( "The analysis for the coin faced up is completed.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Next we analyse what will happen, when the coin is crowned up.\r
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document.write( "2.  With the probability of 1/2 the coin is crowned up.\r\n" );
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document.write( "        Then  with the probability 3/10 2 White balls go from A to B, making B-content equal to (3W,0R),\r\n" );
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document.write( "          OR  with the probability (1/3)*(1/2)  1 White ball and 1 Red ball   go from A to B, making B-content equal to (2W,1R),\r\n" );
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document.write( "          OR  with the probability (1/10)  2 Red balls   go from A to B, making B-content equal to (1W,2R).\r\n" );
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document.write( "    Summary for this step:\r\n" );
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document.write( "            With the probability (1/2)*(3/10) = 3/20 the content of B becomes (3W,0R),        (2a)    and/or\r\n" );
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document.write( "            with the probability (1/2)*(1/3)*(1/2) = 1/12 the content of B becomes (2W,1R),   (2b)    and/or\r\n" );
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document.write( "            with the probability (1/2)*(1/10) = 1/20 the content of B becomes (1W,2R).        (2c)  \r\n" );
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document.write( "     Next step is to withdraw the ball from B. \r\n" );
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document.write( "          In case (2a) we withdraw White ball with the probability 1.\r\n" );
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document.write( "          In case (2b) we withdraw White ball with the probability 2/3.\r\n" );
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document.write( "          In case (2c) we withdraw White ball with the probability 1/3.\r\n" );
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document.write( "     Thus the final probability to have White ball as the final outcome at this consideration is\r\n" );
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document.write( "          +  =  =  = .   (**)\r\n" );
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document.write( "The analysis for the coin crowned up is completed.\r\n" );
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\n" ); document.write( "\n" ); document.write( "The last step to complete the solution is to add (*) and (**) :\r
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document.write( "     +  =  =  =  = 0.6222 = 62.22% (approximately).\r\n" );
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\n" ); document.write( "\n" ); document.write( "Answer. The probability to withdraw white ball is $\"28%2F45\"$ = 0.6222 = 62.22% (approximately).\r
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