document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1086169>Question 1086169</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A game involves selecting a card from a regular 52-card deck and tossing a coin. The coin is a fair coin and is equally likely to land on heads or tails. <BR>\n" );
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document.write( "If the card is a face card, and the coin lands on Heads, you win $8 <BR>\n" );
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document.write( "If the card is a face card, and the coin lands on Tails, you win $2 <BR>\n" );
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document.write( "If the card is not a face card, you lose $2, no matter what the coin shows.\r<BR>\n" );
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document.write( "Question: Find the expected value for this game (expected net gain or loss). (Round your answer to two decimal places.)<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #700336 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=700336\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font color=\"black\" face=\"times\" size=\"3\">Let's define four events:\r<BR>\n" );
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document.write( "F = event of drawing a face card<BR>\n" );
document.write( "N = event of drawing a non-face card<BR>\n" );
document.write( "H = event of the coin landing on heads<BR>\n" );
document.write( "T = event of the coin landing on tails\r<BR>\n" );
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document.write( "The events F and N are complementary. Meaning that one event or the other, but not both, must happen. We either draw a face card (F) or we don't (N). This is why the probabilities add to 1\r<BR>\n" );
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document.write( "P(F) + P(N) = 1\r<BR>\n" );
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document.write( "Solve for P(N) to get\r<BR>\n" );
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document.write( "P(N) = 1 - P(F)\r<BR>\n" );
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document.write( "If we knew the probability of drawing a face card, P(F), then we could find the probability of not getting a face card P(N).\r<BR>\n" );
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document.write( "There are 4 suits with 3 face cards per suit (King, Queen, Jack). So 4*3 = 12 face cards out of 52 cards total.\r<BR>\n" );
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document.write( "Therefore,\r<BR>\n" );
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document.write( "P(F) = probability of drawing a face card<BR>\n" );
document.write( "P(F) = (number of face cards)/(number of cards total)<BR>\n" );
document.write( "P(F) = 12/52<BR>\n" );
document.write( "P(F) = 3/13\r<BR>\n" );
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document.write( "And,\r<BR>\n" );
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document.write( "P(N) = probability of drawing a non-face card<BR>\n" );
document.write( "P(N) = 1-P(F)<BR>\n" );
document.write( "P(N) = 1-3/13<BR>\n" );
document.write( "P(N) = 13/13-3/13<BR>\n" );
document.write( "P(N) = 10/13\r<BR>\n" );
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document.write( "An alternative is to think \"there are 40 non face cards (52-12 = 40) out of 52, so 40/52 = 10/13\"\r<BR>\n" );
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document.write( "Those two probabilities, P(F) and P(N), will be used later. Let's move on to the next two probabilities\r<BR>\n" );
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document.write( "Assuming we have a fair coin (either side is likely to be landed on), this means\r<BR>\n" );
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document.write( "P(H) = 1/2<BR>\n" );
document.write( "P(T) = 1/2\r<BR>\n" );
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document.write( "Note how P(H) + P(T) = 1\r<BR>\n" );
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document.write( "So far we have the four simple probabilities \r<BR>\n" );
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document.write( "P(F) = 3/13, P(N) = 10/13<BR>\n" );
document.write( "P(H) = 1/2, P(T) = 1/2\r<BR>\n" );
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document.write( "Assuming the events of drawing a card and flipping a coin are independent, then we can form the compound probabilities\r<BR>\n" );
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document.write( "P(F & H) = P(F)*P(H) = (3/13)*(1/2) = 3/26<BR>\n" );
document.write( "P(F & T) = P(F)*P(T) = (3/13)*(1/2) = 3/26\r<BR>\n" );
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document.write( "Let's introduce some new notation. Similar to the probability P(X) notation, let's introduce the function V(X) where V is the net value and X is the general event. To be more specific, writing V(F) represents the net value of drawing a face card. \r<BR>\n" );
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document.write( "The three cases we're concerned with are:\r<BR>\n" );
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document.write( "V(F & H) = net value for getting face card and heads = 8<BR>\n" );
document.write( "V(F & T) = net value for getting face card and tails = 2<BR>\n" );
document.write( "V(N) = net value for getting non face card = -2\r<BR>\n" );
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document.write( "The negative value (-2) indicates a loss of 2 dollars.\r<BR>\n" );
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document.write( "When we play the game out, there are three cases:\r<BR>\n" );
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document.write( "Case A = drawing a face card and the coin landing on heads<BR>\n" );
document.write( "Case B = drawing a face card and the coin landing on tails<BR>\n" );
document.write( "Case C = drawing a non-face card\r<BR>\n" );
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document.write( "What we do is multiply the probabilities for each case happening with the net values for each case\r<BR>\n" );
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document.write( "For case A, we have the probability P(F & H) = 3/26 and the net value V(F & H) = 8 so\r<BR>\n" );
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document.write( "P(F & H)*V(F & H) = (3/26)*8 = 24/26 = 12/13\r<BR>\n" );
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document.write( "Similarly for case B\r<BR>\n" );
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document.write( "P(F & T)*V(F & T) = (3/26)*2 = 6/26 = 3/13\r<BR>\n" );
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document.write( "and finally case C\r<BR>\n" );
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document.write( "P(N)*V(N) = (10/13)*(-2) = -20/13\r<BR>\n" );
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document.write( "Once we have those three results (12/13, 3/13, -20/13), we add them up\r<BR>\n" );
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document.write( "(12/13) + (3/13) + (-20/13) = (12+3+(-20))/13<BR>\n" );
document.write( "(12/13) + (3/13) + (-20/13) = -5/13\r<BR>\n" );
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document.write( "The fraction -5/13 converts to the approximate decimal value -0.384615 which rounds to -0.38\r<BR>\n" );
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document.write( "What does this value mean? It is the expected value, ie the expected loss. We expect to lose about 38 cents for each game played. This is not a fair game (because expected value isn't 0). The game clearly favors the house instead of the player.\r<BR>\n" );
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document.write( "Answer: <font color=red>-0.38</font> (represents an average loss of $0.38 or 38 cents per game)</font> </SPAN>\n" );
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