document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1086299>Question 1086299</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The Center for Disease Control, or CDC, has determined that when a person is given a vaccine, the probability that the person will develop immunity to the virus is 0.8. If 4 people are given this vaccine, find the probability that at least 1 of the people will develop immunity to the virus. Give your answer correct to 4 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 #700468 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=700468\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\">Define \"success\" to mean \"developing immunity\"<BR>\n" );
document.write( "X = random discrete variable counting number of successes (takes on the values from X = 0 to X = 4)<BR>\n" );
document.write( "n = 4 is the sample size<BR>\n" );
document.write( "p = 0.8 is the probability of success (probability of developing immunity)\r<BR>\n" );
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document.write( "If we can find P(X = 0), then we can use it to find P(X > 0). Note how P(X = 0) + P(X > 0) = 1. So P(X > 0) = 1 - P(X = 0)\r<BR>\n" );
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document.write( "In this case, k = 0. Also use n = 4 and p = 0.8\r<BR>\n" );
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document.write( "First find the combination value<BR>\n" );
document.write( "n C k = (n!)/(k!*(n-k)!)<BR>\n" );
document.write( "4 C 0 = (4!)/(0!*(4-0)!)<BR>\n" );
document.write( "4 C 0 = (4!)/(0!*4!)<BR>\n" );
document.write( "4 C 0 = (1)/(0!*1)<BR>\n" );
document.write( "4 C 0 = (1)/(1)<BR>\n" );
document.write( "4 C 0 = 1\r<BR>\n" );
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document.write( "Then use it to compute the binomial probability value<BR>\n" );
document.write( "P(X = k) = (n C k)*(p)^(k)*(1-p)^(n-k)<BR>\n" );
document.write( "P(X = 0) = (4 C 0)*(0.8)^(0)*(1-0.8)^(4-0)<BR>\n" );
document.write( "P(X = 0) = (4 C 0)*(0.8)^(0)*(0.2)^(4)<BR>\n" );
document.write( "P(X = 0) = (1)*(0.8)^(0)*(0.2)^4<BR>\n" );
document.write( "P(X = 0) = (1)*(1)*(0.0016)<BR>\n" );
document.write( "P(X = 0) = 0.0016\r<BR>\n" );
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document.write( "We know the value of P(X = 0) now. We can use it to find P(X > 0)\r<BR>\n" );
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document.write( "P(X > 0) = 1 - P(X = 0)<BR>\n" );
document.write( "P(X > 0) = 1 - 0.0016<BR>\n" );
document.write( "P(X > 0) = 0.9984\r<BR>\n" );
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document.write( "Because P(X >= 1) = P(X > 0), we have found the answer. <BR>\n" );
document.write( "The probability that at least one person develops immunity is <font color=red>0.9984</font><BR>\n" );
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