document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1184134>Question 1184134</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider the experiment of throwing two coins.<BR>\n" );
document.write( "a. establish the probability distribution<BR>\n" );
document.write( "b. determine the expected value<BR>\n" );
document.write( "c. determine the standard deviation </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849876 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2264)</B></A><img src=\"/images/stars/stars-13.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=849876\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to analyze the experiment of throwing two coins:\r<BR>\n" );
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document.write( "**a. Probability Distribution:**\r<BR>\n" );
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document.write( "First, let's list the sample space (all possible outcomes):\r<BR>\n" );
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document.write( "* HH (Heads, Heads)<BR>\n" );
document.write( "* HT (Heads, Tails)<BR>\n" );
document.write( "* TH (Tails, Heads)<BR>\n" );
document.write( "* TT (Tails, Tails)\r<BR>\n" );
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document.write( "Since the coins are fair, each outcome has an equal probability of 1/4.\r<BR>\n" );
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document.write( "Let X be the random variable representing the number of heads.  The possible values of X are 0, 1, and 2.  We can now create the probability distribution:\r<BR>\n" );
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document.write( "* P(X = 0) = P(TT) = 1/4<BR>\n" );
document.write( "* P(X = 1) = P(HT) + P(TH) = 1/4 + 1/4 = 2/4 = 1/2<BR>\n" );
document.write( "* P(X = 2) = P(HH) = 1/4\r<BR>\n" );
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document.write( "Here's the probability distribution in table form:\r<BR>\n" );
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document.write( "| X (Number of Heads) | P(X) |<BR>\n" );
document.write( "|---|---|<BR>\n" );
document.write( "| 0 | 1/4 |<BR>\n" );
document.write( "| 1 | 1/2 |<BR>\n" );
document.write( "| 2 | 1/4 |\r<BR>\n" );
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document.write( "**b. Expected Value (E[X] or μ):**\r<BR>\n" );
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document.write( "The expected value is calculated as:\r<BR>\n" );
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document.write( "E[X] = Σ [x * P(X = x)]\r<BR>\n" );
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document.write( "E[X] = (0 * 1/4) + (1 * 1/2) + (2 * 1/4)<BR>\n" );
document.write( "E[X] = 0 + 1/2 + 1/2<BR>\n" );
document.write( "E[X] = 1\r<BR>\n" );
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document.write( "The expected number of heads is 1.\r<BR>\n" );
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document.write( "**c. Standard Deviation (σ):**\r<BR>\n" );
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document.write( "1. **Calculate the variance (Var[X] or σ²):**\r<BR>\n" );
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document.write( "Var[X] = Σ [(x - E[X])² * P(X = x)]\r<BR>\n" );
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document.write( "Var[X] = (0 - 1)² * 1/4 + (1 - 1)² * 1/2 + (2 - 1)² * 1/4<BR>\n" );
document.write( "Var[X] = 1 * 1/4 + 0 * 1/2 + 1 * 1/4<BR>\n" );
document.write( "Var[X] = 1/4 + 0 + 1/4<BR>\n" );
document.write( "Var[X] = 1/2 = 0.5\r<BR>\n" );
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document.write( "2. **Calculate the standard deviation (σ):**\r<BR>\n" );
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document.write( "σ = √Var[X]<BR>\n" );
document.write( "σ = √(1/2)<BR>\n" );
document.write( "σ ≈ 0.707\r<BR>\n" );
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document.write( "The standard deviation of the number of heads is approximately 0.707.<BR>\n" );
document.write( " </SPAN>\n" );
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