Question

if you draw a card with a value of four or less from a standard deck of cards, i will pay you $175 . if not, you pay me $49 . (aces are considered the highest card in the deck.) If you played this game 914 times how much would you expect to win or lose?

Answer 1

Calculating the expected value of the described card game shows that playing 914 times would result in an expected loss of $11,425.

Step-by-step explanation:

To calculate the expected value of playing this card game 914 times, we need to determine the chance of drawing a card with a value of four or less and the outcomes associated with it. In a standard deck of 52 cards, each suit (hearts, spades, clubs, diamonds) has 4 cards that are 4 or less (2, 3, and 4, since aces are high). This gives us a total of 4 suits × 3 cards per suit = 12 cards. So, the probability of drawing one of these cards is 12/52.

The expected value (EV) for a single game is calculated by multiplying the probability of an outcome by the payout or loss for that outcome and summing these values. For this game, EV = (probability of winning) × (payout) + (probability of losing) × (loss). Substituting the values: EV = (12/52) × $175 + (40/52) × (-$49).

After simplifying, we can determine the EV for a single game and then multiply it by 914 to see the expected total value over those games. The calculations would be as follows:

Expected value per game: EV = (12/52) × $175 + (40/52) × (-$49) = ($25) + (-$37.5) = -$12.5

Expected value over 914 games: Total EV = 914 × (-$12.5) = -$11,425

Thus, if you played this game 914 times, you would expect to lose $11,425 in the long run.

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