document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1185185>Question 1185185</A>: <I><SPAN STYLE=\"word-break: break-all;\"> There are 64 cards containing 4 colors: red, yellow, blue, and green (16 cards of each color). The object of the game is to pick a yellow card (player 1) and a blue card (player 2) and pull until you get 8 of that color. The person who reaches 8 of the same color is the winner. No prize. \r<BR>\n" );
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document.write( "1). A card is selected from a deck of 64 cards. There are 4 colors (red, yellow, blue, and green) with 16 cards of each color.\r<BR>\n" );
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document.write( "Use expected value to find the probability of the following:<BR>\n" );
document.write( "a). The card being yellow<BR>\n" );
document.write( "b). The card being blue\r<BR>\n" );
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document.write( "2. a) How would the game be a fair game?<BR>\n" );
document.write( "b) How would the game favor the player?<BR>\n" );
document.write( "c) How would the game favor the creator?<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 #849766 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><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=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=849766\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's the breakdown of the probabilities and how to make the card game fair or favor a player:\r<BR>\n" );
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document.write( "**1. Probability Calculations (Expected Value Approach):**\r<BR>\n" );
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document.write( "While the concept of *expected value* is typically used for calculating the average outcome of a random variable (like the average winnings in a lottery), it's not directly applicable to calculating the probability of drawing a specific color in a single draw.  Probability is calculated directly as:\r<BR>\n" );
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document.write( "Probability = (Number of favorable outcomes) / (Total number of possible outcomes)\r<BR>\n" );
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document.write( "*   **a) Probability of drawing a yellow card:**\r<BR>\n" );
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document.write( "    P(Yellow) = (Number of yellow cards) / (Total number of cards) = 16 / 64 = 1/4 = 0.25\r<BR>\n" );
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document.write( "*   **b) Probability of drawing a blue card:**\r<BR>\n" );
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document.write( "    P(Blue) = (Number of blue cards) / (Total number of cards) = 16 / 64 = 1/4 = 0.25\r<BR>\n" );
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document.write( "**2. Making the Game Fair or Favoring a Player:**\r<BR>\n" );
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document.write( "The current game is *not* fair.  Here's why and how to adjust it:\r<BR>\n" );
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document.write( "*   **a) How to make the game a fair game:**\r<BR>\n" );
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document.write( "    The game is currently biased because Player 1 and Player 2 are looking for different colors.  To make it fair, both players should have the same probability of winning.  Here are a couple of ways:\r<BR>\n" );
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document.write( "    *   **Equal Target Colors:**  Both players should be trying to collect the same color (e.g., both trying to get 8 red cards). This equalizes the probabilities.<BR>\n" );
document.write( "    *   **Adjusted Target Numbers:** If they keep different target colors (yellow and blue), the target number of cards could be adjusted.  For example, since there are 16 of each color, if Player 1 needs 8 yellow cards, Player 2 might only need 8 blue cards to win.\r<BR>\n" );
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document.write( "*   **b) How to favor Player 1 (yellow):**\r<BR>\n" );
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document.write( "    *   **Fewer Yellow Cards Needed:** Player 1 could win by collecting fewer yellow cards (e.g., 7 yellow cards), while Player 2 still needs 8 blue cards.<BR>\n" );
document.write( "    *   **Extra Yellow Cards:**  The deck could be slightly altered to include additional yellow cards (e.g., adding 2 more yellow cards so there are 18 total) and keeping the number of blue cards at 16.<BR>\n" );
document.write( "    *   **Restricted Blue Cards:** Player 2 could be restricted from drawing some of the blue cards.\r<BR>\n" );
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document.write( "*   **c) How to favor the creator:**\r<BR>\n" );
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document.write( "    *   **Hidden Cards:** The creator could \"hide\" some of the blue cards before the game starts, making it harder for Player 2 to reach their target.<BR>\n" );
document.write( "    *   **Unequal Draws:** The players could have a different number of allowed draws. For instance, Player 1 (Yellow) could have more draws than Player 2 (Blue).<BR>\n" );
document.write( "    *   **Biased Deck:** The deck could be secretly altered to contain fewer blue cards.\r<BR>\n" );
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document.write( "It's important to note that any changes to the game to favor one side or the other will make the game unfair.  A fair game requires equal probabilities of winning for all players.<BR>\n" );
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