document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177710>Question 1177710</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A coin is tossed three times. Let Z denote the number of heads on the first toss and W the total number of heads on the 2 tosses. If the coin is unbalanced and a <BR>\n" );
document.write( "head has a 40% chance of occurring, find the joint probability distribution of W and Z.<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 #850402 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=850402\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this problem step-by-step.\r<BR>\n" );
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document.write( "**1. Define Random Variables**\r<BR>\n" );
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document.write( "* **Z:** Number of heads on the first toss (Z can be 0 or 1)<BR>\n" );
document.write( "* **W:** Total number of heads in the last two tosses (W can be 0, 1, or 2)\r<BR>\n" );
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document.write( "**2. Probabilities**\r<BR>\n" );
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document.write( "* Probability of a head (H): P(H) = 0.4<BR>\n" );
document.write( "* Probability of a tail (T): P(T) = 1 - 0.4 = 0.6\r<BR>\n" );
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document.write( "**3. Possible Outcomes and Corresponding W and Z Values**\r<BR>\n" );
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document.write( "Let's list all possible outcomes of three coin tosses and their corresponding W and Z values:\r<BR>\n" );
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document.write( "| Outcome | Z (Heads on 1st Toss) | W (Heads on 2nd & 3rd Tosses) | Probability |<BR>\n" );
document.write( "|---|---|---|---|<BR>\n" );
document.write( "| TTT | 0 | 0 | (0.6)(0.6)(0.6) = 0.216 |<BR>\n" );
document.write( "| TTH | 0 | 1 | (0.6)(0.6)(0.4) = 0.144 |<BR>\n" );
document.write( "| THT | 0 | 1 | (0.6)(0.4)(0.6) = 0.144 |<BR>\n" );
document.write( "| THH | 0 | 2 | (0.6)(0.4)(0.4) = 0.096 |<BR>\n" );
document.write( "| HTT | 1 | 0 | (0.4)(0.6)(0.6) = 0.144 |<BR>\n" );
document.write( "| HTH | 1 | 1 | (0.4)(0.6)(0.4) = 0.096 |<BR>\n" );
document.write( "| HHT | 1 | 1 | (0.4)(0.4)(0.6) = 0.096 |<BR>\n" );
document.write( "| HHH | 1 | 2 | (0.4)(0.4)(0.4) = 0.064 |\r<BR>\n" );
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document.write( "**4. Joint Probability Distribution**\r<BR>\n" );
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document.write( "Now, we create the joint probability distribution table:\r<BR>\n" );
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document.write( "| Z\W | 0 | 1 | 2 |<BR>\n" );
document.write( "|---|---|---|---|<BR>\n" );
document.write( "| 0 | 0.216 | 0.288 (0.144 + 0.144) | 0.096 |<BR>\n" );
document.write( "| 1 | 0.144 | 0.192 (0.096 + 0.096) | 0.064 |\r<BR>\n" );
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document.write( "**Verification**\r<BR>\n" );
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document.write( "Let's verify that the sum of all probabilities is 1:\r<BR>\n" );
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document.write( "0.216 + 0.288 + 0.096 + 0.144 + 0.192 + 0.064 = 1.000\r<BR>\n" );
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document.write( "**Therefore, the joint probability distribution of W and Z is:**\r<BR>\n" );
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document.write( "| Z\W | 0 | 1 | 2 |<BR>\n" );
document.write( "|---|---|---|---|<BR>\n" );
document.write( "| 0 | 0.216 | 0.288 | 0.096 |<BR>\n" );
document.write( "| 1 | 0.144 | 0.192 | 0.064 |<BR>\n" );
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