document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198291>Question 1198291</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Approximately 10.3% of<BR>\n" );
document.write( "American high school students drop out of school before<BR>\n" );
document.write( "graduation. Choose 10 students entering high school at<BR>\n" );
document.write( "random. Find the probability that<BR>\n" );
document.write( "a. No more than two drop out<BR>\n" );
document.write( "b. At least 6 graduate<BR>\n" );
document.write( "c. All 10 stay in school and graduate </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848320 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</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=848320\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a) Probability that no more than two drop out**\r<BR>\n" );
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document.write( "* This means we need to find the probability that 0, 1, or 2 students drop out.\r<BR>\n" );
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document.write( "* We can use the binomial probability formula: <BR>\n" );
document.write( "   P(X = k) = (nCk) * (p^k) * ((1-p)^(n-k)) <BR>\n" );
document.write( "   where:<BR>\n" );
document.write( "      * n = number of trials (10 students)<BR>\n" );
document.write( "      * k = number of successes (dropouts)<BR>\n" );
document.write( "      * p = probability of success (dropout rate = 0.103)<BR>\n" );
document.write( "      * (nCk) = n! / (k! * (n-k)!) \r<BR>\n" );
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document.write( "* P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)\r<BR>\n" );
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document.write( "* Calculate each probability and sum them:<BR>\n" );
document.write( "    * P(X = 0) = (10C0) * (0.103)^0 * (0.897)^10 ≈ 0.3372<BR>\n" );
document.write( "    * P(X = 1) = (10C1) * (0.103)^1 * (0.897)^9 ≈ 0.3708<BR>\n" );
document.write( "    * P(X = 2) = (10C2) * (0.103)^2 * (0.897)^8 ≈ 0.2015\r<BR>\n" );
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document.write( "* P(X ≤ 2) ≈ 0.3372 + 0.3708 + 0.2015 ≈ 0.9095\r<BR>\n" );
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document.write( "**Therefore, the probability that no more than two students drop out is approximately 0.9095.**\r<BR>\n" );
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document.write( "**b) Probability that at least 6 graduate**\r<BR>\n" );
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document.write( "* This means the probability that 4 or fewer students drop out (since 10 - 4 = 6).\r<BR>\n" );
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document.write( "* P(X ≤ 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)\r<BR>\n" );
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document.write( "* Calculate each probability and sum them:<BR>\n" );
document.write( "    * P(X = 3) = (10C3) * (0.103)^3 * (0.897)^7 ≈ 0.0574<BR>\n" );
document.write( "    * P(X = 4) = (10C4) * (0.103)^4 * (0.897)^6 ≈ 0.0118\r<BR>\n" );
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document.write( "* P(X ≤ 4) ≈ 0.9095 + 0.0574 + 0.0118 ≈ 0.9787\r<BR>\n" );
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document.write( "* P(at least 6 graduate) = 1 - P(X ≤ 4) = 1 - 0.9787 ≈ 0.0213\r<BR>\n" );
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document.write( "**Therefore, the probability that at least 6 students graduate is approximately 0.0213.**\r<BR>\n" );
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document.write( "**c) Probability that all 10 stay in school and graduate**\r<BR>\n" );
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document.write( "* This means that 0 students drop out.\r<BR>\n" );
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document.write( "* P(X = 0) = (10C0) * (0.103)^0 * (0.897)^10 ≈ 0.3372\r<BR>\n" );
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document.write( "**Therefore, the probability that all 10 students stay in school and graduate is approximately 0.3372.**<BR>\n" );
document.write( " </SPAN>\n" );
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