document.write( "Question 1155116: There are three boxes containing black and white balls. Box 1 contain 4 white ball, 2 black ball, Box 2 contain 3 white ball and 2 black and Box 3 contain 1 white ball and 2 black. Draw a ball from box 1 and place it in box 2. Then draw a ball from box 2 and place it in box 3. Finally draw a ball from box 3.\r
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Algebra.Com's Answer #777716 by ikleyn(52778) $\"\"$ $\"About$

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\n" ); document.write( "There are three boxes containing black and white balls.
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\n" ); document.write( "Draw a ball from box 1 and place it in box 2.
\n" ); document.write( "Then draw a ball from box 2 and place it in box 3.
\n" ); document.write( "Finally draw a ball from box 3.\r
\n" ); document.write( "\n" ); document.write( "A. What is the probability that the last ball, drawn from box 3 is white?
\n" ); document.write( "B. What is the probability that the last ball drawn is black ball?
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document.write( "Consider 1-st draw of the ball from Box1 to Box2.\r\n" );
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document.write( "With the probability 4/6 you move the white ball, and with probability 2/6 you move the black box.\r\n" );
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document.write( "So, after 1-st draw \r\n" );
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document.write( "    a) with probability 4/6 we will have (4W,2B) in Box2, and \r\n" );
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document.write( "    b) with probability 2/6 we will have ((3W,3B) there.\r\n" );
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document.write( "Now we consider 2-nd draw of the ball from Box2 to Box3. \r\n" );
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document.write( "    We consider case a) first. It creates two sub-cases:\r\n" );
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document.write( "      c) with probability 4/6 we move white ball and then we have then (2W,2B) in Box3;\r\n" );
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document.write( "      d) with probability 2/6 we move black ball and then we have then (1W,3B) in Box3;\r\n" );
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document.write( "    Next consider case b). It creates two sub-cases:\r\n" );
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document.write( "      e) with probability 3/6 we move white ball and then we have then (2W,2B) in Box3;\r\n" );
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document.write( "      f) with probability 3/6 we move black ball and then we have then (1W,3B) in Box3.\r\n" );
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document.write( "Now we analyse each and every of these chains of moves from B1 to B2 and then from B2 to B3 with their probabilities \r\n" );
document.write( "to get what we will have at the end of each chain in Box 3.\r\n" );
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document.write( "      The chain a) --> c) gives (2W,2B) in Box3 with probability  = ;   (1)\r\n" );
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document.write( "      The chain a) --> d) gives (1W,3B) in Box3 with probability  = ;    (2)\r\n" );
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document.write( "      The chain b) --> e) gives (2W,2B) in Box3 with probability  = ;    (3)\r\n" );
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document.write( "      The chain b) --> f) gives (1W,3B) in Box3 with probability  = .    (4)\r\n" );
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document.write( "Now the probability to get white ball from B3 is\r\n" );
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document.write( "      P = .\r\n" );
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document.write( "In this formula, the first factor in each addend is (1),(2),(3),(4);  the second factor comes from the content of Box3.\r\n" );
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document.write( "Finally,  P =  =  = .\r\n" );
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document.write( "It is the ANSWER to question (A) of the problem.\r\n" );
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\n" ); document.write( "\n" ); document.write( "So, part (A) is completed.\r
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\n" ); document.write( "\n" ); document.write( "Part (B) is totally similar. I leave it as a useful exercise to you to check your understanding.\r
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\n" ); document.write( "\n" ); document.write( "In any case, you just have this TEMPLATE in front of you.\r
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