document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1026225>Question 1026225</A>: <I><SPAN STYLE=\"word-break: break-all;\"> At a local fundraising, you can purchase a ticket for $5, and select 3 numbers in any order from 1 to 20. The fundraisers also select 3 numbers.<BR>\n" );
document.write( "If you have all 3 winning numbers, then you win $50.<BR>\n" );
document.write( "If you have exactly 2 winning numbers, then you win $10.<BR>\n" );
document.write( "And finally, if you have exactly 1 winning number, then you win $5.\r<BR>\n" );
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document.write( "What is the  probability of net gain in reduced fraction for  3 winning numbers, and 2 winning numbers? \r<BR>\n" );
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document.write( "Can someone help me set up this problem? I was able to find the amount of net gain but not the probability of net gain. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #641500 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=641500\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> assuming that you can pick the same number 3 times, then the probability of picking the right number for each guess is the same for all 3 guesses.\r<BR>\n" );
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document.write( "for example:\r<BR>\n" );
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document.write( "333 is a winning number.\r<BR>\n" );
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document.write( "under that scenario, then you have a binomial probability type situation.\r<BR>\n" );
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document.write( "the formula for that is:\r<BR>\n" );
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document.write( "p(x) = c(n,x) * p^x * q^(n-x).\r<BR>\n" );
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document.write( "x is the number of times you guessed the right number.<BR>\n" );
document.write( "p is the probability of guessing the right number each time.<BR>\n" );
document.write( "q is the probability of not guessing the right number each time.<BR>\n" );
document.write( "c(n,x) is the number of ways you can guess right x times out of n guesses.\r<BR>\n" );
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document.write( "the probability of guessing a number from 1 to 20 is equal to 1/20 = .05\r<BR>\n" );
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document.write( "the probability of not guessing a number from 1 to 20 is equal to 19/20 = .95.\r<BR>\n" );
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document.write( "the c(n,x) formula means n! / (x! * (n-x)!)\r<BR>\n" );
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document.write( "n is equal to 3 because there are 3 guesses.\r<BR>\n" );
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document.write( "x is equal to 0,1,2,3 because you can guess right 0 times, 1 time, 2 times, or 3 times.\r<BR>\n" );
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document.write( "c(n,x) is the number of times you can get 0 right out of 3, 1 right out of 3, 2 right out of 3, or 3 right out of 3.\r<BR>\n" );
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document.write( "using this formula, you get:\r<BR>\n" );
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document.write( "c(3,0) = 1<BR>\n" );
document.write( "c(3,1) = 3<BR>\n" );
document.write( "c(3,2) = 3<BR>\n" );
document.write( "c(3,3) = 1\r<BR>\n" );
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document.write( "the binomial formula gets you:\r<BR>\n" );
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document.write( "p(0) = c(3,0) * p^0 * q^3 = 1 * .05^0 * .95^3 = 0.857375<BR>\n" );
document.write( "p(1) = c(3,1) * p^1 * q^2 = 3 * .05^1 * .95^2 = 0.135375<BR>\n" );
document.write( "p(2) = c(3,2) * p^2 * q^1 = 3 * .05^2 * .95^1 = 0.007125<BR>\n" );
document.write( "p(3) = c(3,3) * p^3 * q^0 = 1 * .05^3 * .95^0 = 0.000125\r<BR>\n" );
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document.write( "add up all the probabilities and they equal 1, as they should.\r<BR>\n" );
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document.write( "the probability of getting all 3 numbers right is .000125.\r<BR>\n" );
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document.write( "the probability of getting exactly 2 numbers right is .007125\r<BR>\n" );
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document.write( "you buy the ticket so you lost 5 dollars, right off the bat.\r<BR>\n" );
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document.write( "if you get 3 right, then you win 50 dollars, but that only happens .000125 of the time.\r<BR>\n" );
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document.write( "your expected value of getting all 3 right is therefore -5 + .000125 * 50 = -4.99375 dollars.\r<BR>\n" );
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document.write( "if you get 2 right, then you win 10 dollars, but that only happens .007125 of the time.\r<BR>\n" );
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document.write( "your expected value of getting exactly 2 right is therefore -5 + .007125 * 10 = -4.92875.\r<BR>\n" );
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document.write( "what does this mean.\r<BR>\n" );
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document.write( "if you played the game 1 million times, you could expect to pay 1 million * 5 = 5 million dollars.\r<BR>\n" );
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document.write( "you would win 50 dollars .000125 * 1 million = 125 times.\r<BR>\n" );
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document.write( "you would therefore win 125 * 50 = 6250 dollars.\r<BR>\n" );
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document.write( "you paid 5 million dollars and you won 6250 dollars.\r<BR>\n" );
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document.write( "you would have lost 4,993,750 dollars.\r<BR>\n" );
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document.write( "divide that by 1 million and it comes out to losing an average of 4.99375 on each ticket.\r<BR>\n" );
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document.write( "that's what the expected value calculation gave you.\r<BR>\n" );
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document.write( "-4.99375 expected value means you lost 4.99375 dollars on each ticket, assuming that you played the game an infinite number of times.\r<BR>\n" );
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document.write( "i chose 1 million, because that gave me an integer result.\r<BR>\n" );
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document.write( "your total expected value would be:\r<BR>\n" );
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document.write( "-5 + .135375 * 1 + .007125 * 10 + .000125 * 50 = -4.787125\r<BR>\n" );
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document.write( " </SPAN>\n" );
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