document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1196684>Question 1196684</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 1)	There are 4 golden coins and 8 iron coins in a bag. You select one coin from the bag, if it is a golden coin, you keep it; but if it is an iron coin, you put it back in the bag. Find the probability of earning exactly 2 golden coins after:  <BR>\n" );
document.write( "a)	Two consecutive selections<BR>\n" );
document.write( "b)	Three consecutive selections <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 #829637 by </B><A HREF=/tutors/aboutme.mpl?userid=greenestamps><B>greenestamps(13200)</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=829637\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <br><BR>\n" );
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document.write( "**** NOTE ****<br><BR>\n" );
document.write( "My solution to part (b) is different than that by tutor @ikleyn.  Probably her answer is correct.  I read part (b) to mean that exactly 2 gold coins are obtained in EXACTLY 3 selections -- but the statement of the problem doesn't say that.<br><BR>\n" );
document.write( "So look at the solutions using both interpretations to see how they are different.  They both show good mathematical methods; but they solve different problems.<br><BR>\n" );
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document.write( "Note that each time a gold coin is selected it is not returned to the bag, so the number of coins left in the bag decreases by 1; but when an iron coin is selected it is returned to the bag, so the number of coins remaining in the bag does not decrease.<br><BR>\n" );
document.write( "(a) probability of 2 gold coins after 2 selections<br><BR>\n" );
document.write( "Both selections must be gold.<br><BR>\n" );
document.write( "P(gold,gold) = (4/12)(3/11) = 1/11<br><BR>\n" );
document.write( "ANSWER: 1/11<br><BR>\n" );
document.write( "(b) probability of 2 gold coins after exactly 3 selections<br><BR>\n" );
document.write( "The third selection has to be gold; the first two can be either gold then iron or iron then gold.<br><BR>\n" );
document.write( "P(gold,iron,gold) = (4/12)(8/11)(3/11) = 8/121<BR>\n" );
document.write( "P(iron,gold,gold) = (8/12)(4/12)(3/11) = 2/33<br><BR>\n" );
document.write( "The probability of 2 gold on exactly 3 selections is<br><BR>\n" );
document.write( "8/121 + 2/33 = 24/363+22/363 = 46/363<br><BR>\n" );
document.write( "ANSWER: 46/363<br><BR>\n" );
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