document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1168409>Question 1168409</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 73% of all Americans live in cities with population greater than 100,000 people. If 35 Americans are randomly selected, find the probability that\r<BR>\n" );
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document.write( "a. Exactly 26 of them live in cities with population greater than 100,000 people. \r<BR>\n" );
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document.write( "b. At most 26 of them live in cities with population greater than 100,000 people. \r<BR>\n" );
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document.write( "c. At least 25 of them live in cities with population greater than 100,000 people. \r<BR>\n" );
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document.write( "d. Between 21 and 25 (including 21 and 25) of them live in cities with population greater than 100,000 people. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851573 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=851573\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem using the binomial probability distribution.\r<BR>\n" );
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document.write( "**Given:**\r<BR>\n" );
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document.write( "* Probability of an American living in a city with population > 100,000 (p) = 73% = 0.73<BR>\n" );
document.write( "* Number of Americans selected (n) = 35\r<BR>\n" );
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document.write( "**a) Exactly 26 of them live in cities with population greater than 100,000 people.**\r<BR>\n" );
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document.write( "We want to find P(X = 26), where X is the number of Americans living in cities with population > 100,000.\r<BR>\n" );
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document.write( "The binomial probability formula is:\r<BR>\n" );
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document.write( "P(X = k) = (nCk) * p^k * (1-p)^(n-k)\r<BR>\n" );
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document.write( "Where:\r<BR>\n" );
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document.write( "* nCk = n! / (k! * (n-k)!)<BR>\n" );
document.write( "* n = 35<BR>\n" );
document.write( "* k = 26<BR>\n" );
document.write( "* p = 0.73\r<BR>\n" );
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document.write( "P(X = 26) = (35C26) * (0.73)^26 * (0.27)^9\r<BR>\n" );
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document.write( "* 35C26 = 35! / (26! * 9!) = 54,835,425<BR>\n" );
document.write( "* (0.73)^26 ≈ 0.000304<BR>\n" );
document.write( "* (0.27)^9 ≈ 0.000001966\r<BR>\n" );
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document.write( "P(X = 26) = 54,835,425 * 0.000304 * 0.000001966<BR>\n" );
document.write( "P(X = 26) ≈ 0.0327\r<BR>\n" );
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document.write( "**b) At most 26 of them live in cities with population greater than 100,000 people.**\r<BR>\n" );
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document.write( "We want to find P(X ≤ 26), which is the sum of P(X = k) for k = 0 to 26.\r<BR>\n" );
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document.write( "P(X ≤ 26) = Σ [ (35Ck) * (0.73)^k * (0.27)^(35-k) ] for k = 0 to 26\r<BR>\n" );
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document.write( "This is tedious to calculate manually. We can use a calculator or software.\r<BR>\n" );
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document.write( "P(X ≤ 26) ≈ 0.449\r<BR>\n" );
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document.write( "**c) At least 25 of them live in cities with population greater than 100,000 people.**\r<BR>\n" );
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document.write( "We want to find P(X ≥ 25), which is the sum of P(X = k) for k = 25 to 35.\r<BR>\n" );
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document.write( "P(X ≥ 25) = Σ [ (35Ck) * (0.73)^k * (0.27)^(35-k) ] for k = 25 to 35\r<BR>\n" );
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document.write( "Using a calculator or software:\r<BR>\n" );
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document.write( "P(X ≥ 25) ≈ 0.141\r<BR>\n" );
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document.write( "**d) Between 21 and 25 (including 21 and 25) of them live in cities with population greater than 100,000 people.**\r<BR>\n" );
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document.write( "We want to find P(21 ≤ X ≤ 25), which is the sum of P(X = k) for k = 21 to 25.\r<BR>\n" );
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document.write( "P(21 ≤ X ≤ 25) = Σ [ (35Ck) * (0.73)^k * (0.27)^(35-k) ] for k = 21 to 25\r<BR>\n" );
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document.write( "Using a calculator or software:\r<BR>\n" );
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document.write( "P(21 ≤ X ≤ 25) ≈ 0.297\r<BR>\n" );
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document.write( "**Answers:**\r<BR>\n" );
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document.write( "a) Approximately 0.0327<BR>\n" );
document.write( "b) Approximately 0.449<BR>\n" );
document.write( "c) Approximately 0.141<BR>\n" );
document.write( "d) Approximately 0.297<BR>\n" );
document.write( " </SPAN>\n" );
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