document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193441>Question 1193441</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A shipment of 10 items has two defective and eight non defective units. In the inspection of the shipment , a sample of units will be selected and tested. If a defective unit is found , the shipment of 10 units will be rejected.If a sample of five is selected , what is the probability that the shipment will be rejected? </SPAN>\n" );
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document.write( "Answer: <font color=red>7/9</font>\r<BR>\n" );
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document.write( "Explanation:\r<BR>\n" );
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document.write( "We have n = 10 units and we select a sample size of r = 5<BR>\n" );
document.write( "Let's find out how many ways we can do this.<BR>\n" );
document.write( "Use the nCr combination formula. Order does not matter when selecting the units.<BR>\n" );
document.write( "n C r = (n!)/(r!(n-r)!)<BR>\n" );
document.write( "10 C 5 = (10!)/(5!*(10-5)!)<BR>\n" );
document.write( "10 C 5 = (10!)/(5!*5!)<BR>\n" );
document.write( "10 C 5 = (10*9*8*7*6*5!)/(5!*5!)<BR>\n" );
document.write( "10 C 5 = (10*9*8*7*6)/(5!)<BR>\n" );
document.write( "10 C 5 = (10*9*8*7*6)/(5*4*3*2*1)<BR>\n" );
document.write( "10 C 5 = (30240)/(120)<BR>\n" );
document.write( "10 C 5 = 252\r<BR>\n" );
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document.write( "There are 252 ways to select five items from the pool of ten. There may or may not be defective units in this sample of five.<BR>\n" );
document.write( "Let A = 252 so we can use it later.\r<BR>\n" );
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document.write( "Now let's only consider the 8 non-defective units<BR>\n" );
document.write( "We have n = 8 items to pick from and r = 5 selections<BR>\n" );
document.write( "n C r = (n!)/(r!(n-r)!)<BR>\n" );
document.write( "8 C 5 = (8!)/(5!*(8-5)!)<BR>\n" );
document.write( "8 C 5 = (8!)/(5!*3!)<BR>\n" );
document.write( "8 C 5 = (8*7*6*5!)/(5!*3!)<BR>\n" );
document.write( "8 C 5 = (8*7*6)/(3!)<BR>\n" );
document.write( "8 C 5 = (8*7*6)/(3*2*1)<BR>\n" );
document.write( "8 C 5 = (336)/(6)<BR>\n" );
document.write( "8 C 5 = 56<BR>\n" );
document.write( "There are 56 ways to pick a sample of five items, none of which are defective<BR>\n" );
document.write( "Let B = 56\r<BR>\n" );
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document.write( "Subtract the values of A and B<BR>\n" );
document.write( "A-B = 252-56 = 196\r<BR>\n" );
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document.write( "There are 196 ways to have a shipment contain at least one defective unit.\r<BR>\n" );
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document.write( "This works because<BR>\n" );
document.write( "(number of ways to have no defective units) + (number of ways to have at least one defective) = number total ways<BR>\n" );
document.write( "(B) + (number of ways to have at least one defective) = A<BR>\n" );
document.write( "number of ways to have at least one defective = A-B<BR>\n" );
document.write( "number of ways to have at least one defective = 252-56<BR>\n" );
document.write( "number of ways to have at least one defective = 196\r<BR>\n" );
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document.write( "Here's another way to arrive at the figure of 196\r<BR>\n" );
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document.write( "Consider the case of picking exactly one defective unit. There are 2C1 = 2 ways to pick the defective unit, since we have 2 to pick from and one slot to fill. I'm going to skip the steps showing the nCr combination formula, which I'll let you do.<BR>\n" );
document.write( "We then pick from a pool of the 8 non-defective units to fill the remaining 4 slots<BR>\n" );
document.write( "8C4 = 70 ways to pick the non-defective units.<BR>\n" );
document.write( "Overall there are 2*70 = 140 ways to pick exactly one defective unit and 4 non-defective units.<BR>\n" );
document.write( "Let M = 140 to be used later.\r<BR>\n" );
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document.write( "Now consider the case we select exactly 2 defective units<BR>\n" );
document.write( "2C2 = 1 way to pick the two defective units<BR>\n" );
document.write( "8C3 = 56 ways to pick the three non-defective units<BR>\n" );
document.write( "1*56 = 56 ways to pick two defective and three non-defective<BR>\n" );
document.write( "Let N = 56\r<BR>\n" );
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document.write( "M+N = 140+56 = 196 ways to have a shipment contain at least one defective unit.\r<BR>\n" );
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document.write( "What we then do from here is divide the 196 over 252 which was the number of ways to pick a sample of 5 units (defective or not)<BR>\n" );
document.write( "196/252\r<BR>\n" );
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document.write( "That fraction reduces to 7/9 when you divide both parts by the GCF 28\r<BR>\n" );
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document.write( "This is the probability of getting at least one defective item in a randomly selected sample of five, and thereby making the entire batch rejected. <BR>\n" );
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