document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1091034>Question 1091034</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider a game where there is a bag filled with x marbles, one of which is red.<BR>\n" );
document.write( "Initially a contestant is given 5 picks from the bag and then the picked marble is placed back in the bag.  If the contestant<BR>\n" );
document.write( "at any point draws the red marble from the bag they get 5 more picks from the bag.<BR>\n" );
document.write( "&#61623; What is the probability of getting only 5 picks from the bag?<BR>\n" );
document.write( "&#61623; What is the probability of getting exactly 15 picks from the bag? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #705485 by </B><A HREF=/tutors/aboutme.mpl?userid=mathmate><B>mathmate(429)</B></A><img src=\"/images/stars/stars-10.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=mathmate><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=mathmate><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=705485\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Question:<BR>\n" );
document.write( "Consider a game where there is a bag filled with x marbles, one of which is red.<BR>\n" );
document.write( "Initially a contestant is given 5 picks from the bag and then the picked marble is placed back in the bag.  If the contestant<BR>\n" );
document.write( "at any point draws the red marble from the bag they get 5 more picks from the bag.<BR>\n" );
document.write( "&#61623; What is the probability of getting only 5 picks from the bag?<BR>\n" );
document.write( "&#61623; What is the probability of getting exactly 15 picks from the bag?<BR>\n" );
document.write( " <BR>\n" );
document.write( "Solution:<BR>\n" );
document.write( "Picking WITH replacement, <BR>\n" );
document.write( "P(red)=1/x<BR>\n" );
document.write( "P(not red)=(x-1)/x\r<BR>\n" );
document.write( "\n" );
document.write( "(a) 5 picks from the bag<BR>\n" );
document.write( "That means the person had not picked the red marble, not even once.<BR>\n" );
document.write( "Using the multiplication rule,<BR>\n" );
document.write( "P(no red) = P(not red)^5 = [(x-1)/x]^5\r<BR>\n" );
document.write( "\n" );
document.write( "(b) exactly 15 picks<BR>\n" );
document.write( "Case A: The person picked two red marbles from the first five picks, and no red subsequently.<BR>\n" );
document.write( "P(A)=C(5,2)*(1/x)^2*((x-1)/x)^3 * ((x-1)/x)^10<BR>\n" );
document.write( "=10(x-1)^13/x^15<BR>\n" );
document.write( "Case B: The person picked one red marble from the first five, one red from the next five, and no red subsequently<BR>\n" );
document.write( "P(B)=C(5,1)*(1/x)^4 * C(5,1)*(1/x)^4 * ((x-1)/x)^5<BR>\n" );
document.write( "=25(x-1)^13/x^15\r<BR>\n" );
document.write( "\n" );
document.write( "Total<BR>\n" );
document.write( "P(15)=P(A)+P(B)=35(x-1)^13/x^15.<BR>\n" );
document.write( "Check: probability equals about 9% when 6<=x<=10.<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );