document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1010360>Question 1010360</A>: <I><SPAN STYLE=\"word-break: break-all;\"> You have 18 cookies in a box. 5 are oatmeal, 4 are sugar, 3 are peanut butter, and 6 are butter. Show your calculations in answering the questions. <BR>\n" );
document.write( "A. Suppose you randomly choose a cookie, replace it, and randomly choose another. What is the probability that either the first cookie is butter or the second is oatmeal?<BR>\n" );
document.write( "B. Suppose instead that you randomly choose a cookie, but you do not replace it. You then randomly choose another. What is the probability that either the first cookie is butter or the second cookie is oatmeal. </SPAN>\n" );
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document.write( "Question:<BR>\n" );
document.write( "You have 18 cookies in a box. 5 are oatmeal, 4 are sugar, 3 are peanut butter, and 6 are butter. Show your calculations in answering the questions. <BR>\n" );
document.write( "A. Suppose you randomly choose a cookie, replace it, and randomly choose another. What is the probability that either the first cookie is butter or the second is oatmeal?<BR>\n" );
document.write( "B. Suppose instead that you randomly choose a cookie, but you do not replace it. You then randomly choose another. What is the probability that either the first cookie is butter or the second cookie is oatmeal.<BR>\n" );
document.write( " <BR>\n" );
document.write( "Solution:<BR>\n" );
document.write( " <BR>\n" );
document.write( "Given: 18 cookies, 5 oatmeal (O), 4 sugar (S), 3 peanut butter (P) and 6 butter (B).<BR>\n" );
document.write( "Let X be the event of drawing any other cookie.\r<BR>\n" );
document.write( "\n" );
document.write( "A. P(BX or XO) with replacement<BR>\n" );
document.write( "Use a contingency table (multiplied by 18^2=324).<BR>\n" );
document.write( "   B  O  X<BR>\n" );
document.write( "B 36 30 42<BR>\n" );
document.write( "O 30 25 35<BR>\n" );
document.write( "X 42 35 49\r<BR>\n" );
document.write( "\n" );
document.write( "The entries are obtained by <BR>\n" );
document.write( "P(BB)=(6/18)(6/18)=36/324<BR>\n" );
document.write( "P(BO)=(6/18)(5/18)=30/324<BR>\n" );
document.write( "and so on.\r<BR>\n" );
document.write( "\n" );
document.write( "P(BX or XO)<BR>\n" );
document.write( "=(36+30+42)/324+(30+25+35)/324-30/324  (P(BO) was counted twice)<BR>\n" );
document.write( "=14/27<BR>\n" );
document.write( "alternatively,<BR>\n" );
document.write( "P(BX or XO)<BR>\n" );
document.write( "=P(B)+P(O)-P(BO)<BR>\n" );
document.write( "=6/18+5/18-30/324<BR>\n" );
document.write( "=14/27\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "B. P(BX or XO) Without replacement<BR>\n" );
document.write( "Again, use a contingency table (multiplied by 18*17=306).<BR>\n" );
document.write( "   B  O  X<BR>\n" );
document.write( "B 30 30 42<BR>\n" );
document.write( "O 30 20 35<BR>\n" );
document.write( "X 42 35 42<BR>\n" );
document.write( "The above table is obtained by<BR>\n" );
document.write( "P(BB)=(6/18)(5/17)=30/306<BR>\n" );
document.write( "P(BO)=(6/18)(5/17)=30/306<BR>\n" );
document.write( "...<BR>\n" );
document.write( "P(XX)=(7/18)(6/17)=42/306<BR>\n" );
document.write( "So<BR>\n" );
document.write( "P(BX or XO)<BR>\n" );
document.write( "=(30+30+42)/306+(30+20+35)/306-30/306<BR>\n" );
document.write( "=(102+85-30)/306<BR>\n" );
document.write( "=157/306\r<BR>\n" );
document.write( "\n" );
document.write( "Alternatively, <BR>\n" );
document.write( "P(BX or XO)<BR>\n" );
document.write( "=P(B)+P(OO)+P(XO)<BR>\n" );
document.write( "=6/18+(5/18)(4/17)+(7/18)(5/17)<BR>\n" );
document.write( "=157/306 </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );