document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1029561>Question 1029561</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Could you please help me with this question?<BR>\n" );
document.write( "Cal’s Coffee prints prize coupons under the rims of 20% of its paper cups. If you buy ten cups of coffee.<BR>\n" );
document.write( "What is the probability that you would win at least seven prizes?<BR>\n" );
document.write( "What is your expected number of prizes?<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #644575 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=644575\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\">  <BR>\n" );
document.write( "Question:<BR>\n" );
document.write( "Could you please help me with this question?<BR>\n" );
document.write( "Cal’s Coffee prints prize coupons under the rims of 20% of its paper cups. If you buy ten cups of coffee.<BR>\n" );
document.write( "(a) What is the probability that you would win at least seven prizes?<BR>\n" );
document.write( "(b) What is your expected number of prizes?<BR>\n" );
document.write( " <BR>\n" );
document.write( "Solution:<BR>\n" );
document.write( "Here the probability of winning is constant at 20%, and assuming the ten cups of coffee are purchased at random and independently, then we can apply the binomial distribution, given by:<BR>\n" );
document.write( "P(x,n,p)=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%28n%2Cx%29%2Ap%5Ex%2A%281-p%29%5E%28n-x%29 ALIGN=MIDDLE  ALT=\"C%28n%2Cx%29%2Ap%5Ex%2A%281-p%29%5E%28n-x%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "where C(n,x) is the number of combinations of x objects taken out of n,<BR>\n" );
document.write( "p=probability of success <BR>\n" );
document.write( "n=number of objects in the sample.<BR>\n" );
document.write( " <BR>\n" );
document.write( "(a) probability of winning at least 7 prizes<BR>\n" );
document.write( "P(7&le;x&le;10,10,0.2)<BR>\n" );
document.write( "=P(7,0,0.2)+P(8,0,0.2)+P(9,0,0.2)+P(10,0,0.2)<BR>\n" );
document.write( "=<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2810%2C7%29%2A0.2%5E7%2A%281-0.2%29%5E3 ALIGN=MIDDLE  ALT=\"C%2810%2C7%29%2A0.2%5E7%2A%281-0.2%29%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >+<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2810%2C8%29%2A0.2%5E8%2A%281-0.2%29%5E2 ALIGN=MIDDLE  ALT=\"C%2810%2C8%29%2A0.2%5E8%2A%281-0.2%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >+<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2810%2C9%29%2A0.2%5E9%2A%281-0.2%29%5E1 ALIGN=MIDDLE  ALT=\"C%2810%2C9%29%2A0.2%5E9%2A%281-0.2%29%5E1\"   DISABLED_style= \"max-width:90%; height:auto;\" >+<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=C%2810%2C10%29%2A0.2%5E10%2A%281-0.2%29%5E0 ALIGN=MIDDLE  ALT=\"C%2810%2C10%29%2A0.2%5E10%2A%281-0.2%29%5E0\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "=0.00078643+0.00007373+0.00000410+0.00000010<BR>\n" );
document.write( "=0.0008644 (to 7 figures)\r<BR>\n" );
document.write( "\n" );
document.write( "(b) expected value in a binomial distribution is np, which means the expected number of prizes is n*p=10*0.2=2 prizes.<BR>\n" );
document.write( " <BR>\n" );
document.write( "You can read about binomial distributions and more examples at:<BR>\n" );
document.write( "http://www.euclid.host-ed.me/probability/binomialDistribution.html </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );