document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196181>Question 1196181</A>: <I><SPAN STYLE=\"word-break: break-all;\"> There are 15 green apples and 72 red apples in a basket. We keep extracting the apples randomly, one at a time, until we have extracted all 72red apples. What is the probability that the basket is empty when we finish?<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #828908 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13250)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=greenestamps><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=greenestamps><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=828908\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
document.write( "If the basket is empty after we select the last red apple, then the last apple selected is red.<br><BR>\n" );
document.write( "We can represent any sequence of apples selected by a string like \"GGRG...RRGR\" containing 72 \"R\"s and 15 \"G\"s.  Then the probability in question is the probability that the last letter in the string is \"R\".<br><BR>\n" );
document.write( "Since there are 72 red apples and 15 green apples, the probability that the last one selected is red is 72/(72+15) = 72/87.<br><BR>\n" );
document.write( "ANSWER: 72/87<br><BR>\n" );
document.write( "For a more thorough demonstration of why that is the answer, use the basic definition of probability:<br><PRE STYLE=\"white-space: pre-wrap; white-space: -moz-pre-wrap; white-space: -pre-wrap; white-space: -o-pre-wrap; word-wrap: break-word;\">\r\n" );
document.write( "         number of favorable outcomes\r\n" );
document.write( "  P = -----------------------------------\r\n" );
document.write( "       total number of possible outcomes</PRE><BR>\n" );
document.write( "In this problem the possible outcomes are all the possible arrangements of the 72 \"R\"s and 15 \"G\"s, which is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2887%21%29%2F%28%2872%21%29%2815%21%29%29 ALIGN=MIDDLE  ALT=\"%2887%21%29%2F%28%2872%21%29%2815%21%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "And, because the favorable outcomes are those with a red apple last, the number of favorable outcomes is the number of ways of arranging 71 of the 72 red apples and all 15 of the green ones, which is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%2886%21%29%2F%28%2871%21%29%2815%21%29%29 ALIGN=MIDDLE  ALT=\"%2886%21%29%2F%28%2871%21%29%2815%21%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Then the probability in the question is<br><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28%2886%21%29%2F%28%2871%21%29%2815%21%29%29%29%2F%28%2887%21%29%2F%28%2872%21%29%2815%21%29%29%29+=+%2886%21%2F87%21%29%28%28%2872%21%29%2815%21%29%29%2F%28%2871%21%29%2815%21%29%29%29=72%2F87 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
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document.write( "Note added after seeing the response from tutor @ikleyn....<br><BR>\n" );
document.write( "Her math is nearly always very good; but her English is not.  The statement of the problem clearly says that the basket must be empty after removing the 72nd red apple.<br><BR>\n" );
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