document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1194820>Question 1194820</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A game is played as follows: A player places a bet. Then, she rolls a 20-sided die with the numbers<BR>\n" );
document.write( "1  -  20 on it. If the die lands on an even number, the player wins $5. If the die lands on a 1, 5, 7, or 13,<BR>\n" );
document.write( "the player wins $10. Otherwise, the player loses. If it costs $3.50 to bet, what is the expected value of<BR>\n" );
document.write( "this game?<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 #827082 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3816)</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=827082\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Possible Outcomes:<BR>\n" );
document.write( "A = Rolling an even number<BR>\n" );
document.write( "B = Rolling a 1, 5, 7 or 13<BR>\n" );
document.write( "C = Rolling anything else not mentioned by the previous cases\r<BR>\n" );
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document.write( "Here are the various event spaces<BR>\n" );
document.write( "A = {2,4,6,8,10,12,14,16,18,20}<BR>\n" );
document.write( "B = {1,5,7,13}<BR>\n" );
document.write( "C = {3,9,11,15,17,19}\r<BR>\n" );
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document.write( "And here are the corresponding probabilities<BR>\n" );
document.write( "P(A) = 10/20 = 1/2<BR>\n" );
document.write( "P(B) = 4/20 = 1/5<BR>\n" );
document.write( "P(C) = 6/20 = 3/10\r<BR>\n" );
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document.write( "Thing to notice: P(A)+P(B)+P(C) = 1<BR>\n" );
document.write( "The probabilities must sum to 1 in order to have a valid probability distribution. <BR>\n" );
document.write( "Also, the probabilities must be in the interval <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0+%3C=+p+%3C=+1 ALIGN=MIDDLE  ALT=\"0+%3C=+p+%3C=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Let X = net winnings<BR>\n" );
document.write( "This will be the result of subtracting the prize money and the cost to play the game. \r<BR>\n" );
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document.write( "For event A, we have X = 5-3.5 = 1.50<BR>\n" );
document.write( "For event B, we have X = 10-3.5 = 6.50<BR>\n" );
document.write( "For event C, we have X = 0-3.5 = -3.50\r<BR>\n" );
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document.write( "Let's set up a table like so<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >Event</TD><TD >Net Winnings</TD><TD >Probability</TD></tr><tr><TD >A</TD><TD >$1.50</TD><TD >1/2</TD></tr><tr><TD >B</TD><TD >$6.50</TD><TD >1/5</TD></tr><tr><TD >C</TD><TD >-$3.50</TD><TD >3/10</TD></tr>\n" );
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document.write( "Now add on a third column where we multiply the net winnings with their corresponding probability<BR>\n" );
document.write( "For example: 1.5*1/2 = 0.75 in the first row.<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >Event</TD><TD >Net Winnings</TD><TD >Probability</TD><TD >Winnings*Probability</TD></tr><tr><TD >A</TD><TD >$1.50</TD><TD >1/2</TD><TD >0.75</TD></tr><tr><TD >B</TD><TD >$6.50</TD><TD >1/5</TD><TD >1.30</TD></tr><tr><TD >C</TD><TD >-$3.50</TD><TD >3/10</TD><TD >-1.05</TD></tr>\n" );
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document.write( "Adding the results of the third column will get us to the final answer<BR>\n" );
document.write( "0.75 + 1.30 + (-1.05) = 1.00\r<BR>\n" );
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document.write( "Answer: <font color=red>$1.00</font> is the expected winnings.<BR>\n" );
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