Algebra.Com's Answer #846064 by ikleyn(53765)![\"\"](\"/images/stars/stars-13.gif\")  
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document.write( "A bag contain 10 black and 6 white balls. A random sample of 5 balls selected without replacement
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document.write( "is known to contain 3 black balls(Event A). What is the conditional probability that the first
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document.write( "selection is black(Event B) ?
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document.write( " It can be solved/reasoned in more than one way.
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document.write( " I will show you the basic ways to construct the solution.\r
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document.write( " First way\r\n" );
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document.write( "Consider all possible configurations of 5 balls, where three balls are black and the first ball is black.\r\n" );
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document.write( "So, we have two black balls somewhere in two positions from 2 to 5 inclusive, and two white balls in remaining positions.\r\n" );
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document.write( "The number of ways to select two positions from 4 possible positions from 2 to 5 is  =  = 6.\r\n" );
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document.write( "Then two other remaining positions are filled by black balls, so it does not add new configurations.\r\n" );
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document.write( "Thus we have only 6 distinguished configurations for 2 white and 2 black balls in positions from 2 to 5.\r\n" );
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document.write( "From the other side, the number of all possible distinguishable quadruples of white/black balls in 4 position\r\n" );
document.write( "from 2 to 5 is  = 16 (since in each of 4 position we may have black or white ball, independently).\r\n" );
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document.write( "Thus, the probability under the problem's question is P =  =  . ANSWER\r\n" );
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document.write( " Thus, the problem is just solved in this way. \r\n" );
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document.write( " Second way\r\n" );
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document.write( "Another way to think is to determine how many different 4-letter words of two symbols B and W do exist.\r\n" );
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document.write( "This number is  =  = 6. (! Coincides with \"6\" from the solution above !)\r\n" );
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document.write( "In this reasoning, same as in the first reasoning, we use implicitly the fact that in this problem, \r\n" );
document.write( "the pool of given balls (10 black and 6 white) works in the same way as if this pool be unlimited.\r\n" );
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document.write( "Therefore, these concrete numbers \"10 black balls\" and \"6 white balls) do not play any role in the solution.\r\n" );
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document.write( " Third way\r\n" );
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document.write( " P(A and B)\r\n" );
document.write( "Use the formula of the conditional probability P(A|B) = ------------- .\r\n" );
document.write( " P(B)\r\n" );
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document.write( "Here P(B) is the probability to have black ball at the first position and anything in positions from 2 to 5,\r\n" );
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document.write( "which is  =  .\r\n" );
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document.write( "The probability of event (A and B) is the ratio  =  .\r\n" );
document.write( " (Here  is the number of cases (Black or White) in 5 positions).\r\n" );
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document.write( "Thus the final probability in this way is P =  =  = again.\r\n" );
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document.write( "Solved in three different reasoning, for your better understanding.\r
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