document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1199760>Question 1199760</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 84% of owned dogs in the United States are spayed or neutered. Round your answers to four decimal places. If 36 owned dogs are randomly selected, find the probability that\r<BR>\n" );
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document.write( "a. Exactly 31 of them are spayed or neutered. \r<BR>\n" );
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document.write( "b. At most 29 of them are spayed or neutered. \r<BR>\n" );
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document.write( "c. At least 30 of them are spayed or neutered. \r<BR>\n" );
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document.write( "d. Between 29 and 35 (including 29 and 35) of them are spayed or neutered. <BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848282 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=848282\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **a) Probability of exactly 31 spayed/neutered dogs:**\r<BR>\n" );
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document.write( "* This is a binomial probability problem. <BR>\n" );
document.write( "* n = 36 (sample size)<BR>\n" );
document.write( "* p = 0.84 (probability of success - spayed/neutered)<BR>\n" );
document.write( "* q = 1 - p = 0.16 (probability of failure - not spayed/neutered)<BR>\n" );
document.write( "* x = 31 (number of successes)\r<BR>\n" );
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document.write( "* P(X = 31) = (36C31) * (0.84)^31 * (0.16)^(36-31) \r<BR>\n" );
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document.write( "   * where (36C31) is the number of combinations of 36 things taken 31 at a time.\r<BR>\n" );
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document.write( "* Using a binomial probability calculator or statistical software:<BR>\n" );
document.write( "   P(X = 31) ≈ 0.1223\r<BR>\n" );
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document.write( "**b) Probability of at most 29 spayed/neutered dogs:**\r<BR>\n" );
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document.write( "* P(X ≤ 29) = P(X = 0) + P(X = 1) + ... + P(X = 29)\r<BR>\n" );
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document.write( "* Use a cumulative binomial probability calculator or statistical software:<BR>\n" );
document.write( "   P(X ≤ 29) ≈ 0.0419\r<BR>\n" );
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document.write( "**c) Probability of at least 30 spayed/neutered dogs:**\r<BR>\n" );
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document.write( "* P(X ≥ 30) = P(X = 30) + P(X = 31) + ... + P(X = 36)\r<BR>\n" );
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document.write( "* Alternatively:<BR>\n" );
document.write( "   P(X ≥ 30) = 1 - P(X ≤ 29) <BR>\n" );
document.write( "   P(X ≥ 30) = 1 - 0.0419 = 0.9581\r<BR>\n" );
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document.write( "**d) Probability of between 29 and 35 spayed/neutered dogs (inclusive):**\r<BR>\n" );
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document.write( "* P(29 ≤ X ≤ 35) = P(X = 29) + P(X = 30) + ... + P(X = 35)\r<BR>\n" );
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document.write( "* Use a cumulative binomial probability calculator or statistical software to find:<BR>\n" );
document.write( "   P(29 ≤ X ≤ 35) ≈ 0.9954\r<BR>\n" );
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document.write( "**In summary:**\r<BR>\n" );
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document.write( "* a) P(X = 31) ≈ 0.1223<BR>\n" );
document.write( "* b) P(X ≤ 29) ≈ 0.0419<BR>\n" );
document.write( "* c) P(X ≥ 30) ≈ 0.9581<BR>\n" );
document.write( "* d) P(29 ≤ X ≤ 35) ≈ 0.9954<BR>\n" );
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