document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=951522>Question 951522</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The spinner on a wheel of fortune can land with an equal chance on any one of ten regions. Three regions are red, four are blue, two are yellow and one is green. A player wins $4 if the spinner stops on red and $2 if it stops on green. The payer loses $2 if it stop on blue and loses $3 of it stop on yellow. What is the expected value of the game? Please help. I'm confused. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #581150 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=581150\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> expected value = sum (p(x) * v(x)) where sum (p(x)) = 1.<BR>\n" );
document.write( "p(x) is the probability of an event occurring.<BR>\n" );
document.write( "v(x) is the value you get when that event occurs.\r<BR>\n" );
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document.write( "p(red) = .3 and v(x) = 4<BR>\n" );
document.write( "p(blue) = .4 and v(x) = -2<BR>\n" );
document.write( "p(yellow) = .2 and v(x) = -3<BR>\n" );
document.write( "p(green) = .1 and v(x) = 2\r<BR>\n" );
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document.write( "sum of p(x) is equal to 1.\r<BR>\n" );
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document.write( "expected value = sum of p(x) * v(x) = (.3 * 4) - (.4 * 2) - (.2 * 3) + (.1 * 2) = 0\r<BR>\n" );
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document.write( "your expected value is equal to 0.\r<BR>\n" );
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document.write( "this means that, if you played the game many times, you would expect to gain an average of 0 dollars each game.\r<BR>\n" );
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document.write( "consider you played 1000 games and the probability of each occurrence held up.\r<BR>\n" );
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document.write( ".3 * 1000 = 300 * 4 = + 1200<BR>\n" );
document.write( ".4 * 1000 = 400 * -2 = - 800<BR>\n" );
document.write( ".2 * 1000 = 200 * -3 = - 600<BR>\n" );
document.write( ".1 * 1000 = 100 * -2 = + 200\r<BR>\n" );
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document.write( "your net gain would be equal to + 1200 - 800 - 600 + 200 which is equal to 0.\r<BR>\n" );
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document.write( "you neither won or lost any money.\r<BR>\n" );
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document.write( "your average net gain per play would be 0 / 1000 = 0.\r<BR>\n" );
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document.write( "it's easier to see how this works if the expected value isn't 0.\r<BR>\n" );
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document.write( "assume that you lost 2 dollars rather than won 2 dollars when you hit green.\r<BR>\n" );
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document.write( "the probabilities would then be:\r<BR>\n" );
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document.write( "p(red) = .3 and v(x) = 4<BR>\n" );
document.write( "p(blue) = .4 and v(x) = -2<BR>\n" );
document.write( "p(yellow) = .2 and v(x) = -3<BR>\n" );
document.write( "p(green) = .1 and v(x) = -2 *****\r<BR>\n" );
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document.write( "sum of p(x) is still equal to 1.\r<BR>\n" );
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document.write( "expected value = sum of p(x) * v(x) = (.3 * 4) - (.4 * 2) - (.2 * 3) - (.1 * 2) = -.4\r<BR>\n" );
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document.write( "when you lost 2 on hitting green rather than winning 2 when hitting on green, your expected value becomes -.4 dollars per game.\r<BR>\n" );
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document.write( "this means that, if you played the game many times, you would expect to lose an average of 40 cents per game.\r<BR>\n" );
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document.write( "consider you played 1000 games and the probability of each occurrence held up.\r<BR>\n" );
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document.write( ".3 * 1000 = 300 * 4 = + 1200<BR>\n" );
document.write( ".4 * 1000 = 400 * -2 = - 800<BR>\n" );
document.write( ".2 * 1000 = 200 * -3 = - 600<BR>\n" );
document.write( ".1 * 1000 = 100 * -2 = - 200\r<BR>\n" );
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document.write( "your total net gain would be + 1200 - 800 - 600 - 200 which would be equal to - 400.\r<BR>\n" );
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document.write( "that's your total net gain.\r<BR>\n" );
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document.write( "divide that by 1000 times that you played and the average becomes 400 / 1000 = .4 dollars per game.\r<BR>\n" );
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document.write( "you would lose an average of .4 dollars per game which is the same as losing 40 cents per game.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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