Step 1 :Christine has a deck of 10 cards numbered 1 through 10. She is playing a game of chance where she chooses one card from the deck at random. She wins an amount of money equal to the value of the card if an even numbered card is drawn. She loses $7 if an odd numbered card is drawn.
Step 2 :The expected value of a random variable is the long-term average or mean value of the random variable over many independent repetitions of the experiment. It is calculated as the sum of all possible values each multiplied by the probability of its occurrence.
Step 3 :In this case, the possible values are the numbers 1 through 10 and the probabilities are each 1/10 since each card is equally likely to be drawn.
Step 4 :Therefore, the expected value of the game can be calculated as follows: Expected Value = \((1/10 * -7) + (2/10 * 2) + (3/10 * -7) + (4/10 * 4) + (5/10 * -7) + (6/10 * 6) + (7/10 * -7) + (8/10 * 8) + (9/10 * -7) + (10/10 * 10)\)
Step 5 :The expected value of playing the game is \(\boxed{4.5}\) dollars.