document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209734>Question 1209734</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 50 owned dogs are randomly selected, find the probability that\r<BR>\n" );
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document.write( "a. Exactly 44 of them are spayed or neutered. \r<BR>\n" );
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document.write( "b. At most 43 of them are spayed or neutered. \r<BR>\n" );
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document.write( "c. At least 43 of them are spayed or neutered. \r<BR>\n" );
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document.write( "d. Between 37 and 41 (including 37 and 41) 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 #850050 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=850050\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is a binomial probability problem. Here's how to solve it:\r<BR>\n" );
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document.write( "* **n** (number of trials) = 50<BR>\n" );
document.write( "* **p** (probability of success - spayed/neutered) = 0.84<BR>\n" );
document.write( "* **q** (probability of failure - not spayed/neutered) = 1 - p = 0.16\r<BR>\n" );
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document.write( "The binomial probability formula is:  P(x) = (nCx) * p^x * q^(n-x)\r<BR>\n" );
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document.write( "Where nCx represents \"n choose x\" (the binomial coefficient).\r<BR>\n" );
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document.write( "**(a) Exactly 44 are spayed/neutered:**\r<BR>\n" );
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document.write( "P(x = 44) = (50C44) * (0.84)^44 * (0.16)^6<BR>\n" );
document.write( "P(x = 44) = 1,478,745,000 * 0.002011 * 0.00001678<BR>\n" );
document.write( "P(x = 44) ≈ 0.0049\r<BR>\n" );
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document.write( "**(b) At most 43 are spayed/neutered:**\r<BR>\n" );
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document.write( "This means 0 to 43 are spayed/neutered.  It's a cumulative probability.  We can use a binomial cumulative distribution function (CDF) calculator or statistical software for this.  It's the sum of probabilities from x=0 to x=43.\r<BR>\n" );
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document.write( "P(x ≤ 43) ≈ 0.8878 (using a calculator or software)\r<BR>\n" );
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document.write( "**(c) At least 43 are spayed/neutered:**\r<BR>\n" );
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document.write( "This means 43 to 50 are spayed/neutered.  We can use the complement rule:\r<BR>\n" );
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document.write( "P(x ≥ 43) = 1 - P(x < 43) = 1 - P(x ≤ 42)\r<BR>\n" );
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document.write( "Use a binomial CDF calculator:\r<BR>\n" );
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document.write( "P(x ≥ 43) = 1 - 0.8284<BR>\n" );
document.write( "P(x ≥ 43) ≈ 0.1716\r<BR>\n" );
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document.write( "**(d) Between 37 and 41 (inclusive):**\r<BR>\n" );
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document.write( "This means 37, 38, 39, 40, and 41 are spayed/neutered.  We can use the CDF:\r<BR>\n" );
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document.write( "P(37 ≤ x ≤ 41) = P(x ≤ 41) - P(x ≤ 36)\r<BR>\n" );
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document.write( "Use a binomial CDF calculator:<BR>\n" );
document.write( "P(37 ≤ x ≤ 41) = 0.4072 - 0.0150<BR>\n" );
document.write( "P(37 ≤ x ≤ 41) ≈ 0.3922\r<BR>\n" );
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document.write( "**Summary of Answers:**\r<BR>\n" );
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document.write( "* (a) P(x = 44) ≈ 0.0049<BR>\n" );
document.write( "* (b) P(x ≤ 43) ≈ 0.8878<BR>\n" );
document.write( "* (c) P(x ≥ 43) ≈ 0.1716<BR>\n" );
document.write( "* (d) P(37 ≤ x ≤ 41) ≈ 0.3922<BR>\n" );
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