document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198689>Question 1198689</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In the United States, voters who are neither Democrat nor Republican are called Independent. It is believed that 10% of voters are Independent. A survey asked 23 people to identify themselves as Democrat, Republican, or Independent.\r<BR>\n" );
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document.write( "A. What is the probability that none of the people are Independent?<BR>\n" );
document.write( "Probability = ?\r<BR>\n" );
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document.write( "B. What is the probability that fewer than 6 are Independent?<BR>\n" );
document.write( "Probability = ?\r<BR>\n" );
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document.write( "C. What is the probability that more than 2 people are Independent?<BR>\n" );
document.write( "Probability = ? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848241 by </B><A HREF=/tutors/aboutme.mpl?userid=textot><B>textot(100)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=textot><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=848241\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Certainly, let's calculate the probabilities.\r<BR>\n" );
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document.write( "**Assumptions:**\r<BR>\n" );
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document.write( "* The number of Independents follows a binomial distribution.<BR>\n" );
document.write( "* The probability of a voter being Independent is constant (p = 0.10).<BR>\n" );
document.write( "* The voters are independent of each other.\r<BR>\n" );
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document.write( "**A. Probability that none of the people are Independent**\r<BR>\n" );
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document.write( "* In this case, the number of successes (Independent voters) is 0.\r<BR>\n" );
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document.write( "* Using the binomial probability formula:<BR>\n" );
document.write( "   P(X = k) = (nCk) * p^k * (1-p)^(n-k)\r<BR>\n" );
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document.write( "   where:<BR>\n" );
document.write( "      * n = sample size = 23<BR>\n" );
document.write( "      * k = number of successes = 0<BR>\n" );
document.write( "      * p = probability of success (Independent) = 0.10<BR>\n" );
document.write( "      * q = probability of failure (Not Independent) = 1 - p = 0.90\r<BR>\n" );
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document.write( "* P(X = 0) = (23C0) * (0.10)^0 * (0.90)^(23-0) <BR>\n" );
document.write( "             = 1 * 1 * (0.90)^23 <BR>\n" );
document.write( "             ≈ 0.0948\r<BR>\n" );
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document.write( "**Therefore, the probability that none of the people are Independent is approximately 0.0948.**\r<BR>\n" );
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document.write( "**B. Probability that fewer than 6 are Independent**\r<BR>\n" );
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document.write( "* We need to find the probability of 0, 1, 2, 3, 4, or 5 Independents.\r<BR>\n" );
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document.write( "* P(X < 6) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)\r<BR>\n" );
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document.write( "* Calculate each probability using the binomial probability formula and sum them up.\r<BR>\n" );
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document.write( "* Using a calculator or statistical software, we can find:<BR>\n" );
document.write( "   P(X < 6) ≈ 0.9917\r<BR>\n" );
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document.write( "**Therefore, the probability that fewer than 6 people are Independent is approximately 0.9917.**\r<BR>\n" );
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document.write( "**C. Probability that more than 2 people are Independent**\r<BR>\n" );
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document.write( "* We need to find the probability of 3, 4, 5, ..., 23 Independents.\r<BR>\n" );
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document.write( "* P(X > 2) = 1 - P(X ≤ 2) <BR>\n" );
document.write( "             = 1 - [P(X = 0) + P(X = 1) + P(X = 2)]\r<BR>\n" );
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document.write( "* Calculate each probability using the binomial probability formula and sum them up.\r<BR>\n" );
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document.write( "* Using a calculator or statistical software, we can find:<BR>\n" );
document.write( "   P(X > 2) ≈ 0.0593\r<BR>\n" );
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document.write( "**Therefore, the probability that more than 2 people are Independent is approximately 0.0593.**\r<BR>\n" );
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document.write( "I hope this helps! Let me know if you have any other questions.<BR>\n" );
document.write( " </SPAN>\n" );
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