document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1205515>Question 1205515</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider the event of getting a head in 4tosses of a fair coin. Let X be the random variable representing the number of heads minus the number of tails. Create a Probability Distribution table for the random variable X.  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #842427 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=842427\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "The probability distribution is<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD ># heads</TD><TD ># tails</TD><TD >X = heads-tails</TD><TD >P(X)</TD></tr><tr><TD >4</TD><TD >0</TD><TD >4</TD><TD >1/16</TD></tr><tr><TD >3</TD><TD >1</TD><TD >2</TD><TD >4/16</TD></tr><tr><TD >2</TD><TD >2</TD><TD >0</TD><TD >6/16</TD></tr><tr><TD >1</TD><TD >3</TD><TD >-2</TD><TD >4/16</TD></tr><tr><TD >0</TD><TD >4</TD><TD >-4</TD><TD >1/16</TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Below I'll explain how I got each P(X) probability value.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "There are n = 4 tosses of the coin.<BR>\n" );
document.write( "That gives 2^n = 2^4 = 16 different outcomes.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Of those outcomes, there's only one way to get all heads. Same goes for all tails. That explains the 1/16 probability values for the first and last rows.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If there are 3 heads, then there are 4 ways to have this situation. This is because there are 4 places to put the tail. Those 4 outcomes are:<ol><li>HHHT</li><li>HHTH</li><li>HTHH</li><li>THHH</li></ol>Due to symmetry, the same idea applies if there are 3 tails.<BR>\n" );
document.write( "So that's how we get a probability of 4/16 for the 2nd row and 2nd to last row.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "If there are 2 heads, then there are 4C2 = 6 ways to arrange them. The 4C2 refers to the nCr combination formula. Such values can be found in Pascal's Triangle. The 6 ways to have 2 heads and 2 tails are listed here<ol><li>HHTT</li><li>HTTH</li><li>HTHT</li><li>TTHH</li><li>THTH</li><li>THHT</li></ol>So that's how I'm getting 6/16 for the probability of 2 heads.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "I have not reduced the fractions in the P(X) column because I wanted to keep the denominators the same. But if you wanted you could reduce the fractions.<BR>\n" );
document.write( "4/16 = 1/4<BR>\n" );
document.write( "6/16 = 3/8\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Or you can convert all the fractions to decimal form.<BR>\n" );
document.write( "1/16 = 0.0625<BR>\n" );
document.write( "4/16 = 1/4 = 0.25<BR>\n" );
document.write( "6/16 = 3/8 = 0.375<BR>\n" );
document.write( "Each decimal value is exact.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Two things to notice:<ul><li>Each P(X) value is between 0 and 1.</li><li>The P(X) values add to 1.</li></ul><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );