document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1174702>Question 1174702</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Q.No.4. (a) Sixty percent of all buyers of new cars choose automatic transmissions. For a group of five buyers of new cars, calculate the mean and standard deviation for the number of buyers choosing automatic transmissions.  Also calculate the complete distribution.<BR>\n" );
document.write( "(b) What is the probability that a poker hand of 5 cards contain exactly 2 king.<BR>\n" );
document.write( "Q:  If x is a passion random variable with parameter value 1.5, find p(x=1) and P (X = 7). </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850636 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=850636\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down each part of this problem.\r<BR>\n" );
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document.write( "**(a) Automatic Transmissions**\r<BR>\n" );
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document.write( "* **Binomial Distribution:** This situation follows a binomial distribution because:<BR>\n" );
document.write( "    * There are a fixed number of trials (n = 5 buyers).<BR>\n" );
document.write( "    * Each trial has only two outcomes (automatic or not).<BR>\n" );
document.write( "    * The probability of success (automatic) is constant (p = 0.6).<BR>\n" );
document.write( "    * The trials are independent.\r<BR>\n" );
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document.write( "* **Mean (μ):**<BR>\n" );
document.write( "    * μ = np = 5 * 0.6 = 3\r<BR>\n" );
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document.write( "* **Standard Deviation (σ):**<BR>\n" );
document.write( "    * σ = √(np(1-p)) = √(5 * 0.6 * 0.4) = √1.2 ≈ 1.095\r<BR>\n" );
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document.write( "* **Complete Distribution:**<BR>\n" );
document.write( "    * P(X = k) = (n choose k) * p^k * (1-p)^(n-k)<BR>\n" );
document.write( "    * Where (n choose k) = n! / (k! * (n-k)!)\r<BR>\n" );
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document.write( "    * P(X = 0) = (5 choose 0) * (0.6)^0 * (0.4)^5 = 1 * 1 * 0.01024 = 0.01024<BR>\n" );
document.write( "    * P(X = 1) = (5 choose 1) * (0.6)^1 * (0.4)^4 = 5 * 0.6 * 0.0256 = 0.0768<BR>\n" );
document.write( "    * P(X = 2) = (5 choose 2) * (0.6)^2 * (0.4)^3 = 10 * 0.36 * 0.064 = 0.2304<BR>\n" );
document.write( "    * P(X = 3) = (5 choose 3) * (0.6)^3 * (0.4)^2 = 10 * 0.216 * 0.16 = 0.3456<BR>\n" );
document.write( "    * P(X = 4) = (5 choose 4) * (0.6)^4 * (0.4)^1 = 5 * 0.1296 * 0.4 = 0.2592<BR>\n" );
document.write( "    * P(X = 5) = (5 choose 5) * (0.6)^5 * (0.4)^0 = 1 * 0.07776 * 1 = 0.07776\r<BR>\n" );
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document.write( "**(b) Poker Hand with 2 Kings**\r<BR>\n" );
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document.write( "* **Total Possible Hands:** (52 choose 5) = 2,598,960<BR>\n" );
document.write( "* **Ways to Choose 2 Kings:** (4 choose 2) = 6<BR>\n" );
document.write( "* **Ways to Choose 3 Other Cards (not Kings):** (48 choose 3) = 17,296<BR>\n" );
document.write( "* **Probability:** (6 * 17,296) / 2,598,960 ≈ 0.0399\r<BR>\n" );
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document.write( "**(c) Poisson Random Variable**\r<BR>\n" );
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document.write( "* **Poisson Distribution:** X ~ Poisson(λ), where λ = 1.5.<BR>\n" );
document.write( "* **Formula:** P(X = k) = (e^(-λ) * λ^k) / k!\r<BR>\n" );
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document.write( "* **P(X = 1):**<BR>\n" );
document.write( "    * P(X = 1) = (e^(-1.5) * 1.5^1) / 1!<BR>\n" );
document.write( "    * P(X = 1) ≈ (0.2231 * 1.5) / 1 ≈ 0.3347\r<BR>\n" );
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document.write( "* **P(X = 7):**<BR>\n" );
document.write( "    * P(X = 7) = (e^(-1.5) * 1.5^7) / 7!<BR>\n" );
document.write( "    * P(X=7) = (0.2231 * 17.0859)/5040<BR>\n" );
document.write( "    * P(X=7) = 3.811/5040<BR>\n" );
document.write( "    * P(X = 7) ≈ 0.000756<BR>\n" );
document.write( " </SPAN>\n" );
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