Christine has a deck of 10 cards numbered 1 through 10 . She is playing a game of chance.
This game is this: Christine 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.
(a) Find the expected value of playing the game.
dollars
(b) What can Christine expect in the long run, after playing the game many times? (She replaces the card in the deck each time.)
Christine can expect to gain money.
She can expect to win $\square$ dollars per draw.
Christine can expect to lose money.
She can expect to lose $\square$ dollars per draw.
Christine can expect to break even (neither gain nor lose money).

Explanation
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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.