Question 1185185: 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.
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.
Use expected value to find the probability of the following:
a). The card being yellow
b). The card being blue
2. a) How would the game be a fair game?
b) How would the game favor the player?
c) How would the game favor the creator?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's the breakdown of the probabilities and how to make the card game fair or favor a player:
**1. Probability Calculations (Expected Value Approach):**
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:
Probability = (Number of favorable outcomes) / (Total number of possible outcomes)
* **a) Probability of drawing a yellow card:**
P(Yellow) = (Number of yellow cards) / (Total number of cards) = 16 / 64 = 1/4 = 0.25
* **b) Probability of drawing a blue card:**
P(Blue) = (Number of blue cards) / (Total number of cards) = 16 / 64 = 1/4 = 0.25
**2. Making the Game Fair or Favoring a Player:**
The current game is *not* fair. Here's why and how to adjust it:
* **a) How to make the game a fair game:**
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:
* **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.
* **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.
* **b) How to favor Player 1 (yellow):**
* **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.
* **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.
* **Restricted Blue Cards:** Player 2 could be restricted from drawing some of the blue cards.
* **c) How to favor the creator:**
* **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.
* **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).
* **Biased Deck:** The deck could be secretly altered to contain fewer blue cards.
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.
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