document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=480257>Question 480257</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A bag contains 6 red, 4 white and 8 blue balls.  If 3 balls are drawn at random, determine the probability p that at least 1 is red.  \r<BR>\n" );
document.write( "\n" );
document.write( "Here is how I did it:<BR>\n" );
document.write( " <BR>\n" );
document.write( "P(at least 1 Red) = ?<BR>\n" );
document.write( "3 drawings<BR>\n" );
document.write( "Probability that 1 is Red in any trial = (1/18)X6 = 1/3<BR>\n" );
document.write( "Probability that 1 is NOT Red in any trial = q = 1 - p = 1 - 1/3 = 2/3<BR>\n" );
document.write( "Probability that 1 is NOT Red in 3 drawings = (2/3)X(2/3)X(2/3) = 8/27. <BR>\n" );
document.write( "Hence the probability of getting at least 1 Red in 3 drawings = 1-(8/27)= 19/27 = 0.70<BR>\n" );
document.write( "However, the textbook's answer and the solving method are different:<BR>\n" );
document.write( "Probability that none is Red = [(4+8)C3]/18C3 = 55/204 <BR>\n" );
document.write( "Hence the probability that at least 1 is Red = 1-(55/204) = 149/204 = 0.73<BR>\n" );
document.write( "Could you tell me why is my method wrong?  I used the method because a similar problem in my textbook was solved using it. If you like here is the problem I am referring to: <BR>\n" );
document.write( "What is the probability of getting at least 1 ace in 2 throws of a die? \r<BR>\n" );
document.write( "\n" );
document.write( "The probability of getting an ance in any single throw = 1/6. <BR>\n" );
document.write( "The probability of NOT getting an ace in any single throw = 1-(1/6)=5/6.<BR>\n" );
document.write( "The probability of not getting an ace in 2 throws = (5/6)X(5/6)=25/36.<BR>\n" );
document.write( "Hence the probability of getting at least 1 ace in 2 throws = 1-(25/36)=11/36. \r<BR>\n" );
document.write( "\n" );
document.write( "What is the difference between the two types of problems and why my method did not work? \r<BR>\n" );
document.write( "\n" );
document.write( "Thank you so much. \r<BR>\n" );
document.write( "\n" );
document.write( "Have a nice evening. \r<BR>\n" );
document.write( "\n" );
document.write( "Respectfully, \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Ivanka </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #328966 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=328966\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If you were to reach in the bag, pull out 1 ball, check to see if it is red or not, THEN REPLACE IT and repeat the experiment twice more, then your method would produce the correct probability, namely 1 minus the binomial distribution of zero successes in 3 trials where the success of one trial is 1/3.  Notice that you have to do it with replacement because if you don't, the probability of a red changes on each trial (either by reducing the denominator of the single trial probability fraction by one if you DON'T get a red, or reducing both the numerator AND denominator by one if you DO get a red).  Note that the binomial distribution works for the dice game because you are always replacing -- that is you never remove any spots from your dice before rolling them another time.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "However, the experiment described is performed by reaching in the bag, grabbing 3 marbles and checking to see if any of them are red.  Different experiment -- different results.  Here you need the number of ways the single outcome could be zero reds, namely 12 choose 3, divided by the number of ways to draw 3 balls at once, namely 18 choose 3 and then subtract that result from 1.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
document.write( "<div  DISABLED_style= \"text-align:center\"><a  rel=nofollow HREF=\"http://outcampaign.org/\" target=\"_blank\"><IMG STYLE=\"max-width:80%;\" src=\"http://cdn.cloudfiles.mosso.com/c116811/scarlet_A.png\" border=\"0\" alt=\"The Out Campaign: Scarlet Letter of Atheism\" width=\"143\" height=\"122\" /></a></div><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );