document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1203216>Question 1203216</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Please show steps of how to solve the question.\r<BR>\n" );
document.write( "\n" );
document.write( "You are playing a game of chance in which four cards are drawn from a standard deck of 52 cards.\r<BR>\n" );
document.write( "\n" );
document.write( "You guess the suit of each card before it is drawn. The cards are replaced in the deck on each draw.\r<BR>\n" );
document.write( "\n" );
document.write( "You pay $1 to play.<BR>\n" );
document.write( "If you guess the right suit every time, you get your money back and $769.\r<BR>\n" );
document.write( "\n" );
document.write( "Determine your expected profit of playing the game over the long term?\r<BR>\n" );
document.write( "\n" );
document.write( "Expected profit = $\r<BR>\n" );
document.write( "\n" );
document.write( "(round your answer to the nearest dollar) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #838629 by </B><A HREF=/tutors/aboutme.mpl?userid=math_tutor2020><B>math_tutor2020(3817)</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=838629\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Answer: <font color=red size=4>$2.00</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "=================================================================================================\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Explanation:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "X = profit in dollars<BR>\n" );
document.write( "P(X) = probability of getting a certain profit X\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "X takes on exactly two values<BR>\n" );
document.write( "Either X = 769 happens when you get all four guesses correct. <BR>\n" );
document.write( "Or X = -1 to represent losing $1 when getting at least one guess wrong (i.e. one or more guesses wrong)\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "There are 4 suits. The probability of getting a suit correct on any guess is 1/4.<BR>\n" );
document.write( "Getting 4 correct guesses in a row has probability (1/4)^4 = 1/256\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The complement of this is:<BR>\n" );
document.write( "1 - 1/256 = 256/256 - 1/256 = (256-1)/256 = 255/256<BR>\n" );
document.write( "which represents the probability of losing.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "To summarize so far:<BR>\n" );
document.write( "P(X) = 1/256 when X = 769<BR>\n" );
document.write( "P(X) = 255/256 when X = -1\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >X</TD><TD >P(X)</TD></tr><tr><TD >-1</TD><TD >255/256</TD></tr><tr><TD >769</TD><TD >1/256</TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Form a new column labeled X*P(X)<BR>\n" );
document.write( "As the label suggests, we'll multiply each X and P(X) value for any particular row.<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "<tr><TD >X</TD><TD >P(X)</TD><TD >X*P(X)</TD></tr><tr><TD >-1</TD><TD >255/256</TD><TD >-255/256</TD></tr><tr><TD >769</TD><TD >1/256</TD><TD >769/256</TD></tr>\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "E(X) = expected value often denoted as mu = expected profit in this case<BR>\n" );
document.write( "E(X) = sum of X*P(X) values<BR>\n" );
document.write( "E(X) = (-255/256) + (769/256)<BR>\n" );
document.write( "E(X) = (-255+769)/256<BR>\n" );
document.write( "E(X) = 514/256<BR>\n" );
document.write( "E(X) = 2.0078125<BR>\n" );
document.write( "E(X) = <font color=red>2.00</font> when rounding to the nearest dollar.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "When rounding to the nearest cent, we get $2.01 as the average profit.<BR>\n" );
document.write( "Because this profit value is positive, the player should keep playing because the odds are tilted in their favor. \r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Realistically the odds are usually tilted in favor of the house (aka the person(s) who runs the game).<BR>\n" );
document.write( "Otherwise, casino owners would be quickly out of business.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "----------------\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Another approach:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's say the person plays 256 games.<BR>\n" );
document.write( "Each game is defined as \"guess the suit of 4 cards\" (meaning 256 games involves 4*256 = 1024 cards).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "When X = 769, it leads to P(X) = 1/256. <BR>\n" );
document.write( "The player would be expected to win $769 exactly one time out of those 256 attempts.<BR>\n" );
document.write( "The other 255 attempts the player loses $1 each (totaling -255 in losses).<BR>\n" );
document.write( "Wins or losses aren't guaranteed for any particular game. This is just a theoretical thought experiment.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "This helps explain the previous calculation (-255+769)/256 mentioned earlier.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The player would net -255+769 = 514 dollars over those 256 attempts.<BR>\n" );
document.write( "Their average profit would be 514/256 = 2.0078125 dollars which rounds to <font color=red>$2.00</font><BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );