document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196876>Question 1196876</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A student knows the answer to 17 questions out of a list of 20. A test consists of 3 questions, each selected at random from the aforementioned list. To pass the test, the student must get at least 2 correct answers. What chance does the student have of passing the test? </SPAN>\n" );
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document.write( "Let A,B,C represent the 3 questions in which the student does not know the answer to.\r<BR>\n" );
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document.write( "Since the order doesn't matter, there's one way to select those 3 questions. <BR>\n" );
document.write( "Formally you can use the nCr combination formula with n = 3 and r = 3 to get 3C3 = 1.<BR>\n" );
document.write( "Let x = 1 since we'll be using it later.\r<BR>\n" );
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document.write( "Now let's consider the number of ways to select two of the three unknown questions. <BR>\n" );
document.write( "The possibilities are:<BR>\n" );
document.write( "{A,B}<BR>\n" );
document.write( "{A,C}<BR>\n" );
document.write( "{B,C}<BR>\n" );
document.write( "Again, order doesn't matter.<BR>\n" );
document.write( "You can use the nCr combination formula with n = 3 and r = 2 to find that 3C2 = 3\r<BR>\n" );
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document.write( "There are 3 ways to pick exactly 2 of the A,B,C<BR>\n" );
document.write( "Then there are 17 other questions that the student knows<BR>\n" );
document.write( "That makes 3*17 = 51 ways to select 3 questions where 2 are unknown and the third is known<BR>\n" );
document.write( "Let y = 51 since we'll be using it later.\r<BR>\n" );
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document.write( "Let's add the results so far:<BR>\n" );
document.write( "x+y = 1 + 51 = 52<BR>\n" );
document.write( "There are 52 ways for the student to fail the test.<BR>\n" );
document.write( "Either s/he gets all three unknown questions (x = 1) OR the student gets 2 unknown questions (y = 51)\r<BR>\n" );
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document.write( "Use the nCr combination formula to find that there are 17C3 = 680 ways to pick three questions from a pool of 17, where order doesn't matter.\r<BR>\n" );
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document.write( "This means there are 680-52 = 628 ways to pass the test<BR>\n" );
document.write( "This is when the student gets at most 1 unknown question (i.e. 1 or fewer).<BR>\n" );
document.write( "Either the student gets 1 unknown question and 2 known, or the student gets 3 known questions.\r<BR>\n" );
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document.write( "Divide the number of ways to pass over the number of ways to pick the three questions<BR>\n" );
document.write( "628/680 = 0.9235294117647\r<BR>\n" );
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document.write( "Answer: The probability of the student passing is roughly 0.9235<BR>\n" );
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