document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1168408>Question 1168408</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 64% of all violent felons in the prison system are repeat offenders. If 48 violent felons are randomly selected, find the probability that  \r<BR>\n" );
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document.write( "a. Exactly 28 of them are repeat offenders. \r<BR>\n" );
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document.write( "b. At most 30 of them are repeat offenders. \r<BR>\n" );
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document.write( "c. At least 32 of them are repeat offenders. \r<BR>\n" );
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document.write( "d. Between 28 and 34 (including 28 and 34) of them are repeat offenders. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851574 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=851574\">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 a violent felon being a repeat offender (p) = 64% = 0.64<BR>\n" );
document.write( "* Number of violent felons selected (n) = 48\r<BR>\n" );
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document.write( "**a) Exactly 28 of them are repeat offenders.**\r<BR>\n" );
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document.write( "We want to find P(X = 28), where X is the number of repeat offenders.\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 = 48<BR>\n" );
document.write( "* k = 28<BR>\n" );
document.write( "* p = 0.64\r<BR>\n" );
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document.write( "P(X = 28) = (48C28) * (0.64)^28 * (0.36)^20\r<BR>\n" );
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document.write( "* 48C28 = 48! / (28! * 20!) ≈ 1.8396 * 10^13<BR>\n" );
document.write( "* (0.64)^28 ≈ 0.00000216<BR>\n" );
document.write( "* (0.36)^20 ≈ 1.3407 * 10^-9\r<BR>\n" );
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document.write( "P(X = 28) = 1.8396 * 10^13 * 0.00000216 * 1.3407 * 10^-9<BR>\n" );
document.write( "P(X = 28) ≈ 0.0534\r<BR>\n" );
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document.write( "**b) At most 30 of them are repeat offenders.**\r<BR>\n" );
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document.write( "We want to find P(X ≤ 30), which is the sum of P(X = k) for k = 0 to 30.\r<BR>\n" );
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document.write( "P(X ≤ 30) = Σ [ (48Ck) * (0.64)^k * (0.36)^(48-k) ] for k = 0 to 30\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 ≤ 30) ≈ 0.686\r<BR>\n" );
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document.write( "**c) At least 32 of them are repeat offenders.**\r<BR>\n" );
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document.write( "We want to find P(X ≥ 32), which is the sum of P(X = k) for k = 32 to 48.\r<BR>\n" );
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document.write( "P(X ≥ 32) = Σ [ (48Ck) * (0.64)^k * (0.36)^(48-k) ] for k = 32 to 48\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 ≥ 32) ≈ 0.231\r<BR>\n" );
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document.write( "**d) Between 28 and 34 (including 28 and 34) of them are repeat offenders.**\r<BR>\n" );
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document.write( "We want to find P(28 ≤ X ≤ 34), which is the sum of P(X = k) for k = 28 to 34.\r<BR>\n" );
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document.write( "P(28 ≤ X ≤ 34) = Σ [ (48Ck) * (0.64)^k * (0.36)^(48-k) ] for k = 28 to 34\r<BR>\n" );
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document.write( "Using a calculator or software:\r<BR>\n" );
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document.write( "P(28 ≤ X ≤ 34) ≈ 0.669\r<BR>\n" );
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document.write( "**Answers:**\r<BR>\n" );
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document.write( "a) Approximately 0.0534<BR>\n" );
document.write( "b) Approximately 0.686<BR>\n" );
document.write( "c) Approximately 0.231<BR>\n" );
document.write( "d) Approximately 0.669<BR>\n" );
document.write( " </SPAN>\n" );
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