Kyd and North are playing a card game with the following rules:
- Kyd selects a card from a standard 52-card deck.
- If the card is a face card (Jack, Queen, or King), Kyd wins $4 from North.
- If the card is not a face card (which includes the numbered cards from 2 to 10 and the Aces), Kyd pays North $2.
Let's analyze the game in more detail.
Number of Face Cards and Non-Face Cards
In a standard 52-card deck:
-
Face Cards: There are 3 face cards in each suit (Jack, Queen, King), and with 4 suits, that totals to: \[ 3 \text{ face cards per suit} \times 4 \text{ suits} = 12 \text{ face cards} \]
-
Non-Face Cards: The remaining cards are: \[ 52 \text{ total cards} - 12 \text{ face cards} = 40 \text{ non-face cards} \]
Probabilities of Selecting Cards
Now, we can find the probabilities of Kyd drawing a face card or a non-face card.
-
Probability of drawing a face card (P(face)): \[ P(\text{face}) = \frac{\text{Number of face cards}}{\text{Total number of cards}} = \frac{12}{52} = \frac{3}{13} \]
-
Probability of drawing a non-face card (P(non-face)): \[ P(\text{non-face}) = \frac{\text{Number of non-face cards}}{\text{Total number of cards}} = \frac{40}{52} = \frac{10}{13} \]
Expected Value of the Game
The expected value (EV) of Kyd's winnings can be calculated as follows:
-
For drawing a face card:
- Kyd wins $4. The contribution to the expected value is: \[ 4 \cdot P(\text{face}) = 4 \cdot \frac{3}{13} = \frac{12}{13} \]
-
For drawing a non-face card:
- Kyd loses $2 (which means North gains $2). The contribution to the expected value is: \[ -2 \cdot P(\text{non-face}) = -2 \cdot \frac{10}{13} = -\frac{20}{13} \]
Total Expected Value
Now, summing the contributions: \[ EV = \frac{12}{13} + \left(-\frac{20}{13}\right) = \frac{12 - 20}{13} = -\frac{8}{13} \]
Interpretation
The expected value of -\(\frac{8}{13}\) signifies that, on average, Kyd will lose about \(0.615\) dollars per game he plays in the long run. This means that over many games, Kyd can expect to lose money, while North enjoys a favorable game.
3 answers