document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=536397>Question 536397</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I need help on this problem\r<BR>\n" );
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document.write( "There is about an 80% chance that a 20 year old today will be alive at the age of 65. Suppose five 20 year olds were selected at random. What are the following probabilities? <BR>\n" );
document.write( "    a)  P(exactly 3 will be alive at 65)<BR>\n" );
document.write( "    b)  P(at least 3 will be alive at 65)<BR>\n" );
document.write( "    c)  P(at most 2 will be alive at 65)\r<BR>\n" );
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document.write( "Thank you </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #352308 by </B><A HREF=/tutors/aboutme.mpl?userid=fcabanski><B>fcabanski(1391)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=fcabanski><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=352308\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <P><B>If you need help understanding math so you can solve these problems yourself, then one on one online tutoring is the answer ($30/hr).  If you need faster solutions with guaranteed detailed answers, then go with personal problem solving ($3.50-$5.50 per problem).  Contact me at fcabanski@hotmail.com</B><P>\r<BR>\n" );
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document.write( "Probability of an event, P, occurring exactly r times is:<P><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=nCp%2Ap%5Er%2Aq%5E%28n-r%29 ALIGN=MIDDLE  ALT=\"nCp%2Ap%5Er%2Aq%5E%28n-r%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > where n is the number of trials (5 in this case), p is the probability (.8), r the number desired (such as 3 students being alive), and q=1-p which is .2 in this case.<P><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=nCr=n%21%2Fr%21%28n-r%29%21 ALIGN=MIDDLE  ALT=\"nCr=n%21%2Fr%21%28n-r%29%21\"   DISABLED_style= \"max-width:90%; height:auto;\" ><P><BR>\n" );
document.write( "Often we solve such problems by adding a number of probabilities.  For example, if it's the probability that \"at least\" 2 survive, we add the probability for 2, 3, 4 and 5 found with the above formula. <P><BR>\n" );
document.write( "1.  P(exactly 3 alive) gives r =3.  So P = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5C3%2A.8%5E3%2A.2%5E%285-3%29=%285%21%2F3%21%285-3%29%21%29%2A.8%5E3%2A.2%5E2+=+10%2A.512%2A.04+=+.2048 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >= 20.48%<P><BR>\n" );
document.write( "2.  P(at least 3 alive) means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%3E=3 ALIGN=MIDDLE  ALT=\"r%3E=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > so add the P(3) + P(4) + P(5).<P><BR>\n" );
document.write( "P(3)= 20.48%<P><BR>\n" );
document.write( "P(4)= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5C4%2A.8%5E4%2A.2%5E%285-4%29=%285%21%2F4%21%285-4%29%21%29%2A.8%5E4%2A.2%5E1+=+5%2A.4096%2A.2+=+.4096 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > = 40.96%<P><BR>\n" );
document.write( "P(5) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5C5%2A.8%5E5%2A.2%5E%285-5%29=%285%21%2F5%21%285-5%29%21%29%2A.8%5E5%2A.2%5E1+=+1%2A.32768%2A1+=+.32767 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > = 32.77%<P><BR>\n" );
document.write( "P(3 or more) = P(3)+P(4)+P(5)=20.48+40.96+32.77 = 94.21%<P><BR>\n" );
document.write( "3.  P(at most 2)  means <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=r%3C=2 ALIGN=MIDDLE  ALT=\"r%3C=2\"   DISABLED_style= \"max-width:90%; height:auto;\" > so add P(2) + P(1) + P(0).<P><BR>\n" );
document.write( "P(2)= <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5C2%2A.8%5E2%2A.2%5E%285-2%29=%285%21%2F2%21%285-2%29%21%29%2A.8%5E2%2A.2%5E3+=+10%2A.64%2A.008+=+.0512 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >= 5.12%<P><BR>\n" );
document.write( "P(1) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5C1%2A.8%5E1%2A.2%5E%285-1%29=%285%21%2F1%21%285-1%29%21%29%2A.8%5E1%2A.2%5E4+=+5%2A.8%2A.0016+=+.0064 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > = .64%<P><BR>\n" );
document.write( "P(0) = <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5C0%2A.8%5E0%2A.2%5E%285-0%29=%285%21%2F0%21%285-0%29%21%29%2A.8%5E0%2A.2%5E5+=+1%2A.1%2A.00032+=+.00032 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" > = .032%<P><BR>\n" );
document.write( "P(at most 2) = P(2)+P(1)+P(0)= 5.12% + .64% +.032% = 5.79%<P>\r<BR>\n" );
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