document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=118734>Question 118734</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A coin is tossed three times. Use this information to solve these problems.\r<BR>\n" );
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document.write( "1.) List the sample space.\r<BR>\n" );
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document.write( "2.) Find the probability of tossing heads exactly twice.\r<BR>\n" );
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document.write( "3.) Find the probability of tossing tails at least twice.\r<BR>\n" );
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document.write( "Any help would be great! Thanks </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #86928 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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=86928\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let H represent an outcome of heads and T represent an outcome of Tails<BR>\n" );
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document.write( "For three tosses of the coin all the possible outcomes are:<BR>\n" );
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document.write( "H-H-H<BR>\n" );
document.write( "T-H-H<BR>\n" );
document.write( "H-T-H<BR>\n" );
document.write( "H-H-T<BR>\n" );
document.write( "T-H-T<BR>\n" );
document.write( "T-T-H<BR>\n" );
document.write( "H-T-T<BR>\n" );
document.write( "T-T-T<BR>\n" );
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document.write( "These eight possible outcomes are the sample space.<BR>\n" );
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document.write( "To find the probability of tossing heads exactly twice look down the sample space list and<BR>\n" );
document.write( "find any outcome that has exactly two H's. The possibilities are T-H-H, H-T-H, and H-H-T.<BR>\n" );
document.write( "This means that three of the eight possible outcomes contain exactly two heads. Therefore,<BR>\n" );
document.write( "the probability of throwing exactly two heads in three tosses of the coin is 3 out of 8,<BR>\n" );
document.write( "or <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F8 ALIGN=MIDDLE  ALT=\"3%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > or the decimal equivalent of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=3%2F8 ALIGN=MIDDLE  ALT=\"3%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is 0.375 or 37.5 percent.<BR>\n" );
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document.write( "The probability of tossing tails at least twice can be found by looking down the list of eight<BR>\n" );
document.write( "possible outcomes and finding each outcome that has two or more tails in it. The  outcomes that<BR>\n" );
document.write( "have at least two tails in them are T-H-T, T-T-H, H-T-T, and T-T-T. Therefore, there are<BR>\n" );
document.write( "four of the eight outcomes that have two or more tails in them. This means that the probability<BR>\n" );
document.write( "of throwing at least two tails in three tosses is 4 out of 8, which is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=4%2F8 ALIGN=MIDDLE  ALT=\"4%2F8\"   DISABLED_style= \"max-width:90%; height:auto;\" > which reduces<BR>\n" );
document.write( "to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2F2 ALIGN=MIDDLE  ALT=\"1%2F2\"   DISABLED_style= \"max-width:90%; height:auto;\" > and this is 0.50 or 50 percent.<BR>\n" );
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document.write( "Hope this helps you to understand the problem a little better. Note that for each toss<BR>\n" );
document.write( "of a coin there are only two possible outcomes, heads or tails. In three tosses the number<BR>\n" );
document.write( "of possible outcomes is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E3 ALIGN=MIDDLE  ALT=\"2%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" > which equals the eight possible answers that we found. <BR>\n" );
document.write( "If the sample space consisted of tossing the coin 4 times the number of possible outcomes<BR>\n" );
document.write( "would be <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E4 ALIGN=MIDDLE  ALT=\"2%5E4\"   DISABLED_style= \"max-width:90%; height:auto;\" > or 16 possible combinations in the sample space. <BR>\n" );
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document.write( "One way to view the sample space is to raise the number of possible outcomes on each trial<BR>\n" );
document.write( "to the power of the number of trials. Suppose you roll a die twice (the same as rolling<BR>\n" );
document.write( "a pair of dice once). Each die has 6 possible outcomes (the numbers 1, 2, 3, 4, 5, and 6).<BR>\n" );
document.write( "And if you roll the die twice you have <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6%5E2 ALIGN=MIDDLE  ALT=\"6%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > or 36 possible outcomes. Try it as an exercise<BR>\n" );
document.write( "starting with 1&1, 1&2, 1&3, 1&4, 1&5, 1&6, 2&1, 2&2, 2&3, 2&4, and so on until you get<BR>\n" );
document.write( "to the last possible combination of 6&6. If you count up all the possible combinations,<BR>\n" );
document.write( "you will find 36 possible outcomes. <BR>\n" );
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