document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=982730>Question 982730</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello. I desperately need help with this expected value question.<BR>\n" );
document.write( "In the game of chance called \"birdcage\" you have three 6-sided dice. In the<BR>\n" );
document.write( "game you can pick any number from 1 to 6 and the three dice are then “rolled” in a cage. If $1 is bet and exactly one of the number that you picked is rolled then you win $1, if exactly two of the number that you picked are rolled you win $2, and if all three of the dice are the number that you picked you win $10 (in each of these cases you also get your initial $1 bet back). If none of your number wind up being rolled you lose your $1 bet. Suppose that you play 5 games of Birdcage and pick the same number each time.<BR>\n" );
document.write( "a) What is the probability that triples of your number (all three dice come up your numb\r<BR>\n" );
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document.write( "b) What is your total expected win or loss? You answer my indicate both the amount and win or loss. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #603600 by </B><A HREF=/tutors/aboutme.mpl?userid=rothauserc><B>rothauserc(4718)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=rothauserc><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=rothauserc><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=603600\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> P(X=0)=(5/6)^3 = 125/216<BR>\n" );
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document.write( "P(X=1) = P(first dice six) + P(second dice six) + P(third dice six) = (1/6)(5/6)(5/6) + (5/6)(1/6)(5/6) + (5/6)(5/6)(1/6) = 3&#8901;(25/216) = (75/216) = (25/72)<BR>\n" );
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document.write( "P(X=2) = P(first dice not six) + P(second dice not six) + P(third dice not six) = (5/6)(1/6)(1/6) + (1/6)(5/6)(1/6) + (1/6)(1/6)(5/6) = 3&#8901;(5/216) = (15/216) = (5/72)<BR>\n" );
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document.write( "P(X=3) = (1/6)^3 = (1/216)<BR>\n" );
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document.write( "from the above, we have the three probabilities, P(X=1) is probability of getting 1 die with the number you pick, P(X=2) and P(X=3) are the probabilities of getting 2 die with picked number and getting 3 die with picked number<BR>\n" );
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document.write( "a) P(X=3) = 1/216 for one roll<BR>\n" );
document.write( "for 5 rolls it would be (1/216)^5 = 1/470184984576<BR>\n" );
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document.write( "b) expected value for one roll is -1(5/6) + 3(5/72) + 11(1/216) approx &#8722;0.57 which means you lose $.57<BR>\n" );
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document.write( "expected value for 5 rolls is 5(-0.57) = &#8722;2.85 which means you lose $2.85<BR>\n" );
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