document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1180015>Question 1180015</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Suppose 200 people are lined up side-by-side, each one holding a fair coin. Each person flips their coin 64 times; every time it lands heads they step 1 meter forward, each time it lands tails they step 1 meter backward. Use a normal approximation to answer the following question: after everyone finishes their 64 steps, approximately how many people will be standing between 4 and 8 meters behind the starting line? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850128 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850128\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve this problem using a normal approximation:\r<BR>\n" );
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document.write( "**1. Define the Random Variable:**\r<BR>\n" );
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document.write( "* Let X be the number of heads in 64 coin flips.<BR>\n" );
document.write( "* X follows a binomial distribution with n = 64 and p = 0.5.<BR>\n" );
document.write( "* Each person's final position is determined by the difference between the number of heads and tails.\r<BR>\n" );
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document.write( "**2. Relate Position to Heads and Tails:**\r<BR>\n" );
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document.write( "* Let H be the number of heads and T be the number of tails.<BR>\n" );
document.write( "* Final position = H - T<BR>\n" );
document.write( "* We know H + T = 64, so T = 64 - H<BR>\n" );
document.write( "* Final position = H - (64 - H) = 2H - 64\r<BR>\n" );
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document.write( "**3. Determine the Range of Heads:**\r<BR>\n" );
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document.write( "* We want to find the number of people between -8 and -4 meters.<BR>\n" );
document.write( "* -8 ≤ 2H - 64 ≤ -4<BR>\n" );
document.write( "* 56 ≤ 2H ≤ 60<BR>\n" );
document.write( "* 28 ≤ H ≤ 30\r<BR>\n" );
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document.write( "**4. Approximate with a Normal Distribution:**\r<BR>\n" );
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document.write( "* The binomial distribution can be approximated by a normal distribution when n is large enough.<BR>\n" );
document.write( "* Mean (μ) = np = 64 * 0.5 = 32<BR>\n" );
document.write( "* Standard deviation (σ) = √(np(1-p)) = √(64 * 0.5 * 0.5) = √16 = 4\r<BR>\n" );
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document.write( "**5. Calculate Z-scores:**\r<BR>\n" );
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document.write( "* For H = 28: z1 = (28 - 32) / 4 = -1<BR>\n" );
document.write( "* For H = 30: z2 = (30 - 32) / 4 = -0.5\r<BR>\n" );
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document.write( "**6. Find the Probability:**\r<BR>\n" );
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document.write( "* Use a standard normal distribution table or calculator to find the area between z1 and z2.<BR>\n" );
document.write( "* P(-1 ≤ Z ≤ -0.5) = P(Z ≤ -0.5) - P(Z ≤ -1)<BR>\n" );
document.write( "* P(Z ≤ -0.5) ≈ 0.3085<BR>\n" );
document.write( "* P(Z ≤ -1) ≈ 0.1587<BR>\n" );
document.write( "* P(-1 ≤ Z ≤ -0.5) ≈ 0.3085 - 0.1587 = 0.1498\r<BR>\n" );
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document.write( "**7. Calculate the Number of People:**\r<BR>\n" );
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document.write( "* Multiply the probability by the total number of people: 200 * 0.1498 ≈ 29.96\r<BR>\n" );
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document.write( "**Answer:**\r<BR>\n" );
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document.write( "Approximately 30 people will be standing between 4 and 8 meters behind the starting line.<BR>\n" );
document.write( " </SPAN>\n" );
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