document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=974418>Question 974418</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Two Coins.  Flip two coins 50 times and record the number of times exactly one head was obtained. Determine the empirical probability of flipping exactly one head.  How does this compare to the theoretical probability of flipping a coin and getting heads. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #596330 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=596330\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here are my results when I flipped 2 coins 50 times (H = heads, T = tails)\r<BR>\n" );
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document.write( "Each pair of flips represents a single independent trial. Instead of making one long 50x2 table, I made the table a bit more compact (5 columns, 10 trials per column)\r<BR>\n" );
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document.write( "<tr><th>Trial</th><th>Result</th><th>Trial</th><th>Result</th><th>Trial</th><th>Result</th><th>Trial</th><th>Result</th><th>Trial</th><th>Result</th></tr><tr><TD >1</TD><TD >HH</TD><TD >11</TD><TD >TT</TD><TD >21</TD><TD >TT</TD><TD >31</TD><TD >TH</TD><TD >41</TD><TD >TT</TD></tr><tr><TD >2</TD><TD >TT</TD><TD >12</TD><TD >HH</TD><TD >22</TD><TD >HH</TD><TD >32</TD><TD >HH</TD><TD >42</TD><TD >TT</TD></tr><tr><TD >3</TD><TD >HT</TD><TD >13</TD><TD >HH</TD><TD >23</TD><TD >TH</TD><TD >33</TD><TD >HT</TD><TD >43</TD><TD >TT</TD></tr><tr><TD >4</TD><TD >TH</TD><TD >14</TD><TD >TH</TD><TD >24</TD><TD >TT</TD><TD >34</TD><TD >TH</TD><TD >44</TD><TD >HT</TD></tr><tr><TD >5</TD><TD >TT</TD><TD >15</TD><TD >TH</TD><TD >25</TD><TD >HT</TD><TD >35</TD><TD >HH</TD><TD >45</TD><TD >HH</TD></tr><tr><TD >6</TD><TD >TH</TD><TD >16</TD><TD >TH</TD><TD >26</TD><TD >TH</TD><TD >36</TD><TD >HH</TD><TD >46</TD><TD >TT</TD></tr><tr><TD >7</TD><TD >TT</TD><TD >17</TD><TD >HT</TD><TD >27</TD><TD >TT</TD><TD >37</TD><TD >TT</TD><TD >47</TD><TD >TT</TD></tr><tr><TD >8</TD><TD >HH</TD><TD >18</TD><TD >HH</TD><TD >28</TD><TD >TH</TD><TD >38</TD><TD >HH</TD><TD >48</TD><TD >HH</TD></tr><tr><TD >9</TD><TD >HH</TD><TD >19</TD><TD >HT</TD><TD >29</TD><TD >HH</TD><TD >39</TD><TD >TT</TD><TD >49</TD><TD >TH</TD></tr><tr><TD >10</TD><TD >TH</TD><TD >20</TD><TD >TT</TD><TD >30</TD><TD >TH</TD><TD >40</TD><TD >TH</TD><TD >50</TD><TD >HT</TD></tr>\n" );
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document.write( "In <font color=\"red\">red</font> are the trials where exactly one head pops up (either HT or TH)\r<BR>\n" );
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document.write( "<tr><th>Trial</th><th>Result</th><th>Trial</th><th>Result</th><th>Trial</th><th>Result</th><th>Trial</th><th>Result</th><th>Trial</th><th>Result</th></tr><tr><TD >1</TD><TD >HH</TD><TD >11</TD><TD >TT</TD><TD >21</TD><TD >TT</TD><TD >31</TD><TD ><font color=\"red\">TH</font></TD><TD >41</TD><TD >TT</TD></tr><tr><TD >2</TD><TD >TT</TD><TD >12</TD><TD >HH</TD><TD >22</TD><TD >HH</TD><TD >32</TD><TD >HH</TD><TD >42</TD><TD >TT</TD></tr><tr><TD >3</TD><TD ><font color=\"red\">HT</font></TD><TD >13</TD><TD >HH</TD><TD >23</TD><TD ><font color=\"red\">TH</font></TD><TD >33</TD><TD ><font color=\"red\">HT</font></TD><TD >43</TD><TD >TT</TD></tr><tr><TD >4</TD><TD ><font color=\"red\">TH</font></TD><TD >14</TD><TD ><font color=\"red\">TH</font></TD><TD >24</TD><TD >TT</TD><TD >34</TD><TD ><font color=\"red\">TH</font></TD><TD >44</TD><TD ><font color=\"red\">HT</font></TD></tr><tr><TD >5</TD><TD >TT</TD><TD >15</TD><TD ><font color=\"red\">TH</font></TD><TD >25</TD><TD ><font color=\"red\">HT</font></TD><TD >35</TD><TD >HH</TD><TD >45</TD><TD >HH</TD></tr><tr><TD >6</TD><TD ><font color=\"red\">TH</font></TD><TD >16</TD><TD ><font color=\"red\">TH</font></TD><TD >26</TD><TD ><font color=\"red\">TH</font></TD><TD >36</TD><TD >HH</TD><TD >46</TD><TD >TT</TD></tr><tr><TD >7</TD><TD >TT</TD><TD >17</TD><TD ><font color=\"red\">HT</font></TD><TD >27</TD><TD >TT</TD><TD >37</TD><TD >TT</TD><TD >47</TD><TD >TT</TD></tr><tr><TD >8</TD><TD >HH</TD><TD >18</TD><TD >HH</TD><TD >28</TD><TD ><font color=\"red\">TH</font></TD><TD >38</TD><TD >HH</TD><TD >48</TD><TD >HH</TD></tr><tr><TD >9</TD><TD >HH</TD><TD >19</TD><TD ><font color=\"red\">HT</font></TD><TD >29</TD><TD >HH</TD><TD >39</TD><TD >TT</TD><TD >49</TD><TD ><font color=\"red\">TH</font></TD></tr><tr><TD >10</TD><TD ><font color=\"red\">TH</font></TD><TD >20</TD><TD >TT</TD><TD >30</TD><TD ><font color=\"red\">TH</font></TD><TD >40</TD><TD ><font color=\"red\">TH</font></TD><TD >50</TD><TD ><font color=\"red\">HT</font></TD></tr>\n" );
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document.write( "There are 21 trials with a red HT or TH in them. The rest are either HH or TT.\r<BR>\n" );
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document.write( "There are 50 trials total. So the empirical probability is 21/50 = 0.42 which is pretty close to the theoretical probability of 1/2 = 0.5\r<BR>\n" );
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document.write( "Jim\r<BR>\n" );
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document.write( "------------------------------------------------------------------------------------------------------------------------ </SPAN>\n" );
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