document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178055>Question 1178055</A>: <I><SPAN STYLE=\"word-break: break-all;\"> An experiment consists of drawing four objects from a container, which holds eight operable, six defective, and 10 semioperable objects. Let X be the number of operable objects drawn and Y the number of defective objects drawn.\r<BR>\n" );
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document.write( "(a) Find the joint probability function of the bivariate random variable (X, Y).<BR>\n" );
document.write( "(b) Find P(X = 3, Y = 0).<BR>\n" );
document.write( "(c) Find P(X < 3, Y = 1).<BR>\n" );
document.write( "(d) Give a graphical presentation of (a), (b), and (c).\r<BR>\n" );
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document.write( "Thank you :) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850370 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</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=850370\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step-by-step.\r<BR>\n" );
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document.write( "**Understanding the Problem**\r<BR>\n" );
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document.write( "* **Total Objects:** 8 (operable) + 6 (defective) + 10 (semioperable) = 24<BR>\n" );
document.write( "* **Objects Drawn:** 4<BR>\n" );
document.write( "* **X:** Number of operable objects drawn<BR>\n" );
document.write( "* **Y:** Number of defective objects drawn\r<BR>\n" );
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document.write( "**a. Joint Probability Function P(X = x, Y = y)**\r<BR>\n" );
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document.write( "We need to find the probability of drawing x operable objects and y defective objects, given that we draw 4 objects in total.\r<BR>\n" );
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document.write( "* **Number of Semioperable Objects Drawn:** 4 - x - y<BR>\n" );
document.write( "* **Total Combinations:** C(24, 4) = 24! / (4! * 20!) = 10626<BR>\n" );
document.write( "* **Combinations of Operable Objects:** C(8, x)<BR>\n" );
document.write( "* **Combinations of Defective Objects:** C(6, y)<BR>\n" );
document.write( "* **Combinations of Semioperable Objects:** C(10, 4 - x - y)\r<BR>\n" );
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document.write( "The joint probability function is:\r<BR>\n" );
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document.write( "* P(X = x, Y = y) = [C(8, x) * C(6, y) * C(10, 4 - x - y)] / C(24, 4)\r<BR>\n" );
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document.write( "Where:\r<BR>\n" );
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document.write( "* x can be 0, 1, 2, 3, 4<BR>\n" );
document.write( "* y can be 0, 1, 2, 3, 4<BR>\n" );
document.write( "* 0 ≤ x + y ≤ 4\r<BR>\n" );
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document.write( "**b. P(X = 3, Y = 0)**\r<BR>\n" );
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document.write( "* P(X = 3, Y = 0) = [C(8, 3) * C(6, 0) * C(10, 1)] / C(24, 4)<BR>\n" );
document.write( "* C(8, 3) = 8! / (3! * 5!) = 56<BR>\n" );
document.write( "* C(6, 0) = 1<BR>\n" );
document.write( "* C(10, 1) = 10<BR>\n" );
document.write( "* P(X = 3, Y = 0) = (56 * 1 * 10) / 10626 = 560 / 10626 ≈ 0.0527\r<BR>\n" );
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document.write( "**c. P(X < 3, Y = 1)**\r<BR>\n" );
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document.write( "We need to calculate P(X = 0, Y = 1) + P(X = 1, Y = 1) + P(X = 2, Y = 1).\r<BR>\n" );
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document.write( "* **P(X = 0, Y = 1):** [C(8, 0) * C(6, 1) * C(10, 3)] / C(24, 4) = (1 * 6 * 120) / 10626 = 720 / 10626 ≈ 0.0678<BR>\n" );
document.write( "* **P(X = 1, Y = 1):** [C(8, 1) * C(6, 1) * C(10, 2)] / C(24, 4) = (8 * 6 * 45) / 10626 = 2160 / 10626 ≈ 0.2033<BR>\n" );
document.write( "* **P(X = 2, Y = 1):** [C(8, 2) * C(6, 1) * C(10, 1)] / C(24, 4) = (28 * 6 * 10) / 10626 = 1680 / 10626 ≈ 0.1581\r<BR>\n" );
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document.write( "* **P(X < 3, Y = 1) = 0.0678 + 0.2033 + 0.1581 = 0.4292**\r<BR>\n" );
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document.write( "**d. Graphical Presentation**\r<BR>\n" );
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document.write( "**(a) Joint Probability Function Table**\r<BR>\n" );
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document.write( "We can create a table showing the probabilities for various values of X and Y.\r<BR>\n" );
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document.write( "| X\Y | 0        | 1        | 2        | 3        | 4        |<BR>\n" );
document.write( "|-----|----------|----------|----------|----------|----------|<BR>\n" );
document.write( "| 0   | 0.0099   | 0.0678   | 0.0283   | 0.0022   | 0        |<BR>\n" );
document.write( "| 1   | 0.0475   | 0.2033   | 0.0847   | 0.0071   | 0        |<BR>\n" );
document.write( "| 2   | 0.1188   | 0.1581   | 0.0494   | 0.0035   | 0        |<BR>\n" );
document.write( "| 3   | 0.0527   | 0.0413   | 0.0098   | 0.0003   | 0        |<BR>\n" );
document.write( "| 4   | 0.0028   | 0.0011   | 0.0001   | 0        | 0        |\r<BR>\n" );
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document.write( "**(b) P(X = 3, Y = 0)**\r<BR>\n" );
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document.write( "This is a single point on the graph. It's a bar at (3, 0) with height 0.0527.\r<BR>\n" );
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document.write( "**(c) P(X < 3, Y = 1)**\r<BR>\n" );
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document.write( "This involves summing the probabilities for (0, 1), (1, 1), and (2, 1). We can represent this graphically as a sum of bars at these points.\r<BR>\n" );
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document.write( "**Graphical Representation Notes:**\r<BR>\n" );
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document.write( "* For the joint probability function, you'd typically use a 3D bar chart or a heatmap to visualize the probabilities for different combinations of X and Y.<BR>\n" );
document.write( "* For (b) and (c), you could use a bar chart or simply mark the points on a graph.<BR>\n" );
document.write( "* Due to the complexity of the data, a software like python matplotlib or R is recommended for the graphical presentation.<BR>\n" );
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