document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1206285>Question 1206285</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An urn contains 5 red balls and 8 yellow balls. If Vince chooses 10 balls at random from the urn, what is the probability that he will select 4 red balls and 6  yellow balls? Round your answer to 3 decimal places. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #843609 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=843609\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> the set contains 5 red and 8 yellow balls.\r<BR>\n" );
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document.write( "i think this can be done this way.<BR>\n" );
document.write( "p(4 red and 6 yellow) = c(5,4) * c(8,6) / c(13,10) = .4895104895.\r<BR>\n" );
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document.write( "i think it can also be done this way.<BR>\n" );
document.write( "p(4 red and 6 yellow) = 5/13 * 4/12 * 3/11 * 2/10 * 8/9 * 7/8 * 6/7 * 5/6 * 4/5 * 3/4 * c(10,4) = .4895104895.\r<BR>\n" );
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document.write( "the first way is essentially the number of ways you can get your want divided by the total number of possible ways.\r<BR>\n" );
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document.write( "the formula is p(4 red and 6 yellow) = c(5,4) * c(8,6) / c(13,10)\r<BR>\n" );
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document.write( "that translates to:\r<BR>\n" );
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document.write( "the number of ways you can get 4 red balls from 5 red balls is c(5,4).<BR>\n" );
document.write( "the number of ways you can get 6 yellow balls from 8 yellow balls is c(8,6).<BR>\n" );
document.write( "the number of ways you can get 4 red balls and 6 yellow balls becomes c(5,4) * c(8,6) which translates to the number of ways you can get 4 red balls times the number of ways you can get 6 yellow balls.<BR>\n" );
document.write( "that's the numerator.<BR>\n" );
document.write( "the number of possible ways to get 10 balls out of 13 balls is c(13,10).<BR>\n" );
document.write( "that's the denominator.\r<BR>\n" );
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document.write( "the second way looks at the probability of getting them in a specified order.\r<BR>\n" );
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document.write( "that formula becomes p(4 red and 6 yellow) = 5/13 * 4/12 * 3/11 * 2/10 * 8/9 * 7/8 * 6/7 * 5/6 * 4/5 * 3/4 * c(10,4) = .4895104895.\r<BR>\n" );
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document.write( "on your first draw you pick a red.<BR>\n" );
document.write( "the probability of that is 5/13.<BR>\n" );
document.write( "on your second draw you pick another red.<BR>\n" );
document.write( "the probability of that is 4/12.<BR>\n" );
document.write( "ditto until you pick the 4th ball as red.<BR>\n" );
document.write( "starting from the 5th ball you draw, you are not looking at yellow.<BR>\n" );
document.write( "probability of the firth ball being yellow is 8/9.<BR>\n" );
document.write( "that's because there are 8 yellow balls left out of a total of 9 left.<BR>\n" );
document.write( "this continues until your last draw of a yellow ball whose probability is 3/4.<BR>\n" );
document.write( "there are 4 balls left.<BR>\n" );
document.write( "they are RYYY<BR>\n" );
document.write( "that's because you took our 4 red and 5 yellow for a total of 9<BR>\n" );
document.write( "13 minus 9 = 4 balls left<BR>\n" );
document.write( "5 red minus 4 red = 1 red left<BR>\n" );
document.write( "8 yellow minus 5 yellow = 3 yellow left.\r<BR>\n" );
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document.write( "you wind up with 10 balls that were picked in a certain order.<BR>\n" );
document.write( "the number of ways those 10 could be ordered is c(10,4).<BR>\n" );
document.write( "that's the number of possible ways you can arrange 4 red balls from 10 total balls.<BR>\n" );
document.write( "the other number of ways those 10 could be ordered is c(10,6).<BR>\n" );
document.write( "that's the number of possible ways you can arrange 6 yellow balls from 10 total balls.<BR>\n" );
document.write( "c(10,4) = 210<BR>\n" );
document.write( "c(10,6) also = 210.\r<BR>\n" );
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document.write( "two different ways get you the same answer which at least confirms the ways are consistent.\r<BR>\n" );
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document.write( "to see how each formula works, you can work a different problems where the numbers are small enough that you can show each possible say separately.<BR>\n" );
document.write( "i did that with this problem and it confirms both methods are good.\r<BR>\n" );
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document.write( "your solution is that the probability is equal to .4895104895.\r<BR>\n" );
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document.write( "round that to 3 decimal places to get .490 which is the same as .49.<BR>\n" );
document.write( "even though .49 is the same as .490, you should probably show it as .490 just to indicate that you rounded to 3 decimal places, and not 2.\r<BR>\n" );
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