document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1013826>Question 1013826</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Pouches of specialty chemicals are tested for presence of impurities a or b, which render the chemical unfit for use.  The fitness for use is tested separately.  Suppose 10% of the pouches have impurity a, 5% have impurity b,1% have both impurities a & b.  Other pouches are free of impurities.  The probabilities of being unfit for use are 1%, 2%, 6%, and 0.1% for pouches with impurity a,impurity b, both impurities a & b, and no impurities respectively. What is the probability of a randomly chosen pouch being found unfit for use? </SPAN>\n" );
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document.write( "Question:<BR>\n" );
document.write( "Pouches of specialty chemicals are tested for presence of impurities a or b, which render the chemical unfit for use.  The fitness for use is tested separately.  Suppose 10% of the pouches have impurity a, 5% have impurity b,1% have both impurities a & b.  Other pouches are free of impurities.  The probabilities of being unfit for use are 1%, 2%, 6%, and 0.1% for pouches with impurity a,impurity b, both impurities a & b, and no impurities respectively. What is the probability of a randomly chosen pouch being found unfit for use?<BR>\n" );
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document.write( "Solution:<BR>\n" );
document.write( "When we randomly select one pouch, the outcomes are:<BR>\n" );
document.write( "A: has impurity a<BR>\n" );
document.write( "B: has impurity b<BR>\n" );
document.write( "AB: has both impurities a and b<BR>\n" );
document.write( "N: has neither impurity.<BR>\n" );
document.write( " <BR>\n" );
document.write( "We are given <BR>\n" );
document.write( "P(AB)=0.01<BR>\n" );
document.write( "P(A or AB)=0.10<BR>\n" );
document.write( "P(B or AB)=0.05<BR>\n" );
document.write( "We conclude that<BR>\n" );
document.write( "P(A)=P(A or AB)-P(AB)=0.10-0.01=0.09<BR>\n" );
document.write( "P(B)=P(B or AB)-P(B)=0.05-0.01=0.04<BR>\n" );
document.write( "Hence<BR>\n" );
document.write( "P(N)=1-(P(A)+P(B)+P(AB)=1-(0.09+0.04+0.01)=1-0.14=0.86<BR>\n" );
document.write( " <BR>\n" );
document.write( "Thus the probability of picking<BR>\n" );
document.write( "P{A,B,AB,N}={0.09,0.04,0.01,0.86}<BR>\n" );
document.write( "Note that the sum of the four probabilities is 1.0 since they are complementary.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Let a, b, ab, n be the events that the chosen pouch is unfit for use, then the corresponding probabilities for unfit pouches are (as given):<BR>\n" );
document.write( "P(a,b,ab,n)={0.01,0.02,0.06,0.001}<BR>\n" );
document.write( "  <BR>\n" );
document.write( "Therefore the probability of ending up with any unfit pouch is the product of the probability of picking the category of pouch and the corresponding unfit probability, or P(A)*P(a)....<BR>\n" );
document.write( "P(A)*P(a)=0.09*0.01=9*10^-4<BR>\n" );
document.write( "P(B)*P(b)=0.04*0.02=8*10^-4<BR>\n" );
document.write( "P(AB)*P(ab)=0.01*0.06=6*10^-4<BR>\n" );
document.write( "P(N)*P(n)=0.85*0.001=8.6*10^-4<BR>\n" );
document.write( " <BR>\n" );
document.write( "Total probability of unfit pouch<BR>\n" );
document.write( "=&Sigma;P(X)*P(x)<BR>\n" );
document.write( "=(9+8+6+8.6)*10^4<BR>\n" );
document.write( "=0.00316<BR>\n" );
document.write( " <BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );