Final answer:
To decide whether to play the card game, calculate the expected value by multiplying the probabilities of winning or losing by their respective amounts. The expected value is approximately $5.385 per game, meaning it is profitable in the long term.
Step-by-step explanation:
To determine whether you should play a game involving selecting cards from a deck of 52 cards, you need to calculate the expected value of the game.
In the game described, you win $30 if you draw a face card and lose $2 if you do not. With 12 face cards in the deck, the probability of drawing a face card is 12/52, and the probability of not drawing a face card is 40/52.
To find the expected value (E), calculate E = (probability of winning) × (amount won) + (probability of losing) × (amount lost). Plugging in values gives us E = (12/52) × 30 + (40/52) × (-2).
Calculating further, E = (12/52) × 30 + (40/52) × (-2) ≈ 6.923 - 1.538 ≈ 5.385. So, on average, you win about $5.38 per game in the long run. However, this calculation does not account for variability or risk; it solely indicates the long-term average outcome.