document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192747>Question 1192747</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A firm which conducts consumer survey by mail has found that 30 percent of those families receiving a questionnaire will return it. In a survey of 10 families, what is the probability that <BR>\n" );
document.write( "(i) exactly five families will return the questionnaire. <BR>\n" );
document.write( "(ii) between 3 to 5 five families (inclusive) will return the questionnaire. (iii) exactly 10 will return the questionnaire. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848952 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=848952\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This problem involves a **binomial distribution**, where the probability of success (a family returning the questionnaire) is \( p = 0.3 \), and the number of trials is \( n = 10 \).\r<BR>\n" );
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document.write( "The probability mass function of a binomial random variable \( X \) is:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "P(X = k) = \binom{n}{k} p^k (1-p)^{n-k},<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "where:<BR>\n" );
document.write( "- \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \) is the binomial coefficient,<BR>\n" );
document.write( "- \( k \) is the number of successes.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### (i) Probability that exactly 5 families return the questionnaire<BR>\n" );
document.write( "Substitute \( n = 10 \), \( k = 5 \), and \( p = 0.3 \):<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "P(X = 5) = \binom{10}{5} (0.3)^5 (0.7)^5.<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### (ii) Probability that between 3 and 5 families (inclusive) return the questionnaire<BR>\n" );
document.write( "This includes the probabilities for \( X = 3 \), \( X = 4 \), and \( X = 5 \):<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "P(3 \leq X \leq 5) = P(X = 3) + P(X = 4) + P(X = 5).<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "For each term:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "P(X = k) = \binom{10}{k} (0.3)^k (0.7)^{10-k}, \quad k = 3, 4, 5.<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "### (iii) Probability that exactly 10 families return the questionnaire<BR>\n" );
document.write( "Substitute \( n = 10 \), \( k = 10 \), and \( p = 0.3 \):<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "P(X = 10) = \binom{10}{10} (0.3)^{10} (0.7)^0.<BR>\n" );
document.write( "\]<BR>\n" );
document.write( "Since \( \binom{10}{10} = 1 \), this simplifies to:<BR>\n" );
document.write( "\[<BR>\n" );
document.write( "P(X = 10) = (0.3)^{10}.<BR>\n" );
document.write( "\]\r<BR>\n" );
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document.write( "Let me compute these probabilities.\r<BR>\n" );
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document.write( "It seems like I can’t do more advanced data analysis right now. Please try again later. If you'd like, I can provide the setup and formula for you to calculate the probabilities manually. Let me know how you'd like to proceed! </SPAN>\n" );
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