document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1200014>Question 1200014</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A simple gambling game involves the throw of a die. Players are charged 40 cents if the die shows 1,2 or 3 they are charged nothing for 4 and they receive 30 cents or 60 cents respectively for 5 or 6. Let the random variable X be the payout to the player.<BR>\n" );
document.write( "Calculate the expected value and how much profit would the casino expect to make on 100 games?\r<BR>\n" );
document.write( "\n" );
document.write( "If the backer weighted the die so that it is 50% more likely to turn up each of the results 1,2,3 as to turn up 4,5 or 6.<BR>\n" );
document.write( "What is the probability now of each outcome?<BR>\n" );
document.write( "Find expected value now. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #834021 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=834021\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=black size=3><BR>\n" );
document.write( "Problem 1: <font color=blue>A simple gambling game involves the throw of a die. <BR>\n" );
document.write( "Players are charged 40 cents if the die shows 1,2 or 3 they are charged nothing for 4 and they receive 30 cents or 60 cents respectively for 5 or 6. <BR>\n" );
document.write( "Let the random variable X be the payout to the player.<BR>\n" );
document.write( "Calculate the expected value and how much profit would the casino expect to make on 100 games?</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "X = payout, in cents, to the player<BR>\n" );
document.write( "X can take on the following values: {-40,0,30,60}<BR>\n" );
document.write( "The negative value is where the player loses money.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The probability of rolling a 1, 2 or 3 is 3/6.<BR>\n" );
document.write( "This is the probability of X = -40 happening.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The probability of rolling a 4 is 1/6 because there's one side labeled \"4\" out of 6 sides total.<BR>\n" );
document.write( "This is when X = 0\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The probability of rolling a 5 is also 1/6.<BR>\n" );
document.write( "This is the probability of X = 30 happening.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The probability of rolling a 6 is also 1/6.<BR>\n" );
document.write( "This is the probability of X = 60 happening.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Let's set up a table with each X and P(X) value.<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "\r\n" );
document.write( "<tr><TD >X</TD><TD >P(X)</TD></tr>\r\n" );
document.write( "<tr><TD >-40</TD><TD >3/6</TD></tr>\r\n" );
document.write( "<tr><TD >0</TD><TD >1/6</TD></tr>\r\n" );
document.write( "<tr><TD >30</TD><TD >1/6</TD></tr>\r\n" );
document.write( "<tr><TD >60</TD><TD >1/6</TD></tr>\r\n" );
document.write( "\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now multiply across the rows to form the X*P(X) column.<BR>\n" );
document.write( "Example: -40 times 3/6 = -120/6<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "\r\n" );
document.write( "<tr><TD >X</TD><TD >P(X)</TD><TD >X*P(X)</TD></tr>\r\n" );
document.write( "<tr><TD >-40</TD><TD >3/6</TD><TD >-120/6</TD></tr>\r\n" );
document.write( "<tr><TD >0</TD><TD >1/6</TD><TD >0/6</TD></tr>\r\n" );
document.write( "<tr><TD >30</TD><TD >1/6</TD><TD >30/6</TD></tr>\r\n" );
document.write( "<tr><TD >60</TD><TD >1/6</TD><TD >60/6</TD></tr>\r\n" );
document.write( "\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Add the items in that third column to get the expected payout.<BR>\n" );
document.write( "(-120/6)+(0/6)+(30/6)+(60/6)<BR>\n" );
document.write( "= (-120+0+30+60)/6<BR>\n" );
document.write( "= -30/6<BR>\n" );
document.write( "= -5\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "A negative payout means the player lost money.<BR>\n" );
document.write( "The player expects to lose, on average, 5 cents per game.<BR>\n" );
document.write( "From the perspective of the casino, they are winning 5 cents per game on average.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Over the course of 100 games, the casino would expect to gain 100*5 = 500 cents = $5<BR>\n" );
document.write( "Or the calculation would be 100*(0.05 dollars) = 5 dollars.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Answers:<BR>\n" );
document.write( "Expected payout per game = -5 cents (perspective of the player)<BR>\n" );
document.write( "Casino profit on one hundred games = $5\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "========================================================\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Problem 2: <font color=blue>If the backer weighted the die so that it is 50% more likely to turn up each of the results 1,2,3 as to turn up 4,5 or 6.<BR>\n" );
document.write( "What is the probability now of each outcome?<BR>\n" );
document.write( "Find expected value now.</font>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "50% more likely involves the multiplier 1.5, which converts to 3/2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Rolling a 1,2, or 3 initially had probability 3/6 aka 1/2.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Increase that by 50% to get (1.5)*(1/2) = (3/2)*(1/2) = 3/4\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Therefore, event X = -40 has probability P(X) = 3/4\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "On this weighted die, rolling a 1,2, or 3 has probability 3/4. <BR>\n" );
document.write( "That means 1-3/4 = 1/4 is the leftover amount for the other values\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Split that remaining bit into three equal pieces: (1/4)*(1/3) = 1/12<BR>\n" );
document.write( "Rolling a 4 has probability 1/12<BR>\n" );
document.write( "Rolling a 5 has probability 1/12<BR>\n" );
document.write( "Rolling a 6 has probability 1/12\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The event X = 0 has probability 1/12<BR>\n" );
document.write( "The event X = 30 has probability 1/12<BR>\n" );
document.write( "The event X = 60 has probability 1/12<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "\r\n" );
document.write( "<tr><TD >X</TD><TD >P(X)</TD></tr>\r\n" );
document.write( "<tr><TD >-40</TD><TD >3/4</TD></tr>\r\n" );
document.write( "<tr><TD >0</TD><TD >1/12</TD></tr>\r\n" );
document.write( "<tr><TD >30</TD><TD >1/12</TD></tr>\r\n" );
document.write( "<tr><TD >60</TD><TD >1/12</TD></tr>\r\n" );
document.write( "\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "For the sake of consistency, let's get each denominator to 12.<BR>\n" );
document.write( "3/4 = (3/4)*(3/3) = 9/12<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "\r\n" );
document.write( "<tr><TD >X</TD><TD >P(X)</TD></tr>\r\n" );
document.write( "<tr><TD >-40</TD><TD >9/12</TD></tr>\r\n" );
document.write( "<tr><TD >0</TD><TD >1/12</TD></tr>\r\n" );
document.write( "<tr><TD >30</TD><TD >1/12</TD></tr>\r\n" );
document.write( "<tr><TD >60</TD><TD >1/12</TD></tr>\r\n" );
document.write( "\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Now form the X*P(X) column<BR>\n" );
document.write( "<TABLE  border = \"1\" cellpadding = \"5\">\n" );
document.write( "\r\n" );
document.write( "<tr><TD >X</TD><TD >P(X)</TD><TD >X*P(X)</TD></tr>\r\n" );
document.write( "<tr><TD >-40</TD><TD >9/12</TD><TD >-360/12</TD></tr>\r\n" );
document.write( "<tr><TD >0</TD><TD >1/12</TD><TD >0/12</TD></tr>\r\n" );
document.write( "<tr><TD >30</TD><TD >1/12</TD><TD >30/12</TD></tr>\r\n" );
document.write( "<tr><TD >60</TD><TD >1/12</TD><TD >60/12</TD></tr>\r\n" );
document.write( "\n" );
document.write( "</TABLE>\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Add the X*P(X) items<BR>\n" );
document.write( "(-360/12)+(0/12)+(30/12)+(60/12)<BR>\n" );
document.write( "= (-360+0+30+60)/12<BR>\n" );
document.write( "= -270/12<BR>\n" );
document.write( "= -22.50\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "The player expects to lose, on average, 22.50 cents per game.<BR>\n" );
document.write( "The casino expects to gain, on average, 22.50 cents per game.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Playing 100 games means the casino expects to make 0.225*100 = 22.50 dollars\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "Answers:<BR>\n" );
document.write( "The probability of each outcome is shown in the table above<BR>\n" );
document.write( "Expected payout for the player = -22.5 cents<BR>\n" );
document.write( "</font> </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );