document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1198592>Question 1198592</A>: <I><SPAN STYLE=\"word-break: break-all;\"> a) A random variable x has a binomial distribution with n = 12. <BR>\n" );
document.write( "Given that p = 0.25, find\r<BR>\n" );
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document.write( "(i) p (x < 5) (From table III or IV of Appendix A)\r<BR>\n" );
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document.write( "(ii) p (x ≥ 7) (From table III or IV of Appendix A)\r<BR>\n" );
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document.write( "b)Given that p(x=0) = 0.05, find the value of p to 3 decimal places.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848414 by </B><A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><B>ElectricPavlov(122)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><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=848414\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a) Binomial Distribution with n = 12 and p = 0.25**\r<BR>\n" );
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document.write( "* **(i) p(x < 5)**\r<BR>\n" );
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document.write( "   * Using a binomial probability table or calculator (like the one provided in the code response), we find: <BR>\n" );
document.write( "      * p(x < 5) = 0.8819\r<BR>\n" );
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document.write( "* **(ii) p(x ≥ 7)**\r<BR>\n" );
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document.write( "   * p(x ≥ 7) = 1 - p(x < 7) <BR>\n" );
document.write( "   * Using a binomial probability table or calculator, we find:<BR>\n" );
document.write( "      * p(x < 7) = 0.9856<BR>\n" );
document.write( "   * Therefore, p(x ≥ 7) = 1 - 0.9856 = 0.0144\r<BR>\n" );
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document.write( "**b) Finding the value of p given p(x=0) = 0.05**\r<BR>\n" );
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document.write( "* We know that for a binomial distribution:<BR>\n" );
document.write( "   * p(x=0) = (nCx) * p^x * (1-p)^(n-x) <BR>\n" );
document.write( "   * where nCx is the binomial coefficient (number of combinations of n items taken x at a time)\r<BR>\n" );
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document.write( "* In this case, n = 12 and x = 0<BR>\n" );
document.write( "   * p(x=0) = (12C0) * p^0 * (1-p)^(12-0) <BR>\n" );
document.write( "   * p(x=0) = 1 * 1 * (1-p)^12 \r<BR>\n" );
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document.write( "* We are given p(x=0) = 0.05<BR>\n" );
document.write( "   * Therefore, 0.05 = (1-p)^12\r<BR>\n" );
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document.write( "* Solve for p:<BR>\n" );
document.write( "   * p = 1 - (0.05)^(1/12) <BR>\n" );
document.write( "   * p ≈ 0.221\r<BR>\n" );
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document.write( "**Therefore:**\r<BR>\n" );
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document.write( "* **a) (i) p(x < 5) = 0.8819**<BR>\n" );
document.write( "   * **(ii) p(x ≥ 7) = 0.0144**<BR>\n" );
document.write( "* **b) The value of p is approximately 0.221**<BR>\n" );
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