Final answer:
The game involves selecting a card from a regular 52-card deck. The expected value for this game is $0, indicating that you will break even in the long run. Therefore, playing this game will not result in winning money, but it will also not result in losing money.
Step-by-step explanation:
The game involves selecting a card from a regular 52-card deck. To find the expected value for this game (expected net gain or loss), we need to consider the probability of each outcome and its associated payoff.
If the card is a face card (there are 12 face cards in a deck), and the coin lands on heads (50% chance), you win $6. So the probability of this event is (12/52) * (1/2).
If the card is a face card, and the coin lands on tails (also 50% chance), you win $2. So the probability of this event is (12/52) * (1/2).
If the card is not a face card (40 non-face cards in a deck), you lose $2. So the probability of this event is (40/52) * 1.
To calculate the expected value, we multiply the probabilities of each outcome by their respective payoff amounts and add them together. The expected value for this game is (12/52) * (1/2) * $6 + (12/52) * (1/2) * $2 + (40/52) * (-$2).
Based on the calculations, the expected net gain or loss for this game is $0. The long-term average profits and losses on this game indicate that, on average, you will break even.
Therefore, playing this game will not result in winning money, but it will also not result in losing money in the long run.
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