document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1025823>Question 1025823</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In problem 2–42, suppose that American investment institutions enter this<BR>\n" );
document.write( "new market, and that their probabilities for successful instruments are<BR>\n" );
document.write( "Goldman Sachs 70%<BR>\n" );
document.write( "Salomon Brothers 82%<BR>\n" );
document.write( "Fidelity 80%<BR>\n" );
document.write( "Smith Barney 90%<BR>\n" );
document.write( "What is the probability that at least one of these four instruments is successful?<BR>\n" );
document.write( "Assume independence. </SPAN>\n" );
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document.write( "Question:<BR>\n" );
document.write( "In problem 2–42, suppose that American investment institutions enter this<BR>\n" );
document.write( "new market, and that their probabilities for successful instruments are<BR>\n" );
document.write( "Goldman Sachs 70%<BR>\n" );
document.write( "Salomon Brothers 82%<BR>\n" );
document.write( "Fidelity 80%<BR>\n" );
document.write( "Smith Barney 90%<BR>\n" );
document.write( "What is the probability that at least one of these four instruments is successful?<BR>\n" );
document.write( "Assume independence.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Solution:<BR>\n" );
document.write( "Here we have 4 (assumed) independent events each with a given probability.<BR>\n" );
document.write( "The rule of multiplication says that the probability of all events will succeed is the product of the respective probabilities.<BR>\n" );
document.write( "Since we are looking for at P(A)=P(\"least one of them will succeed\"), which is the complement of P(B)=P(\"all of them will fail\"), we only need to find P(B) and obtain P(A) by the relation P(A)=1-P(B) whenever A and B are complementary.<BR>\n" );
document.write( " <BR>\n" );
document.write( "To find P(B), we need to define the events<BR>\n" );
document.write( "P(G)=1-0.70=0.30 that Goldman Sachs will <i>fail</i><BR>\n" );
document.write( "P(S)=1-0.82=0.18 that Salomon Brothers will <i>fail</i><BR>\n" );
document.write( "P(F)=1-0.80=0.20 that Fidelity will <i>fail</i>, and<BR>\n" );
document.write( "P(B)=1-0.90=0.10 that Smith Barney will <i>fail</i>.\r<BR>\n" );
document.write( "\n" );
document.write( "The probability that ALL of them will fail is therefore given by the multiplication rule as <BR>\n" );
document.write( "P(B)=P(fail)=P(G)*P(S)*P(F)*P(B),<BR>\n" );
document.write( "and the probability that at least one of the will succeed is<BR>\n" );
document.write( "P(A)=1-P(B), as explained above. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );