Let's break down the game between Kyd and North.

Deck Overview:

• A standard deck contains 52 cards, divided as follows:
  • 13 cards from each of the four suits (Hearts, Diamonds, Clubs, Spades).
  • The face cards (Jack, Queen, King) include 3 from each suit (4 suits), which means there are a total of 12 face cards.
  • The remaining 40 cards are non-face cards (numbered from Ace to 10).

Probabilities:

• The probability of Kyd selecting a face card is: $P(\text{Face card}) = \frac{12}{52} = \frac{3}{13}$
• The probability of Kyd selecting a non-face card is: $P(\text{Non-face card}) = \frac{40}{52} = \frac{10}{13}$

Payoff Analysis:

1. If Kyd draws a face card, North pays Kyd $4.
2. If Kyd draws a non-face card, Kyd pays North $3.

Expected Value for Kyd:

To determine Kyd's expected payoff, we calculate the weighted average of the outcomes based on their probabilities.

$\text{Expected value} = (\text{Payoff for face card} \times P(\text{Face card})) + (\text{Payoff for non-face card} \times P(\text{Non-face card}))$ Substituting the values:

$\text{Expected value} = (4 \times \frac{3}{13}) + (-3 \times \frac{10}{13})$

Now, let's compute that:

$\text{Expected value} = \frac{12}{13} - \frac{30}{13} = \frac{-18}{13} \approx -1.38$

Conclusion:

The expected value of this game for Kyd is approximately -$1.38. This means that on average, Kyd would lose $1.38 per draw in the long run.

Would you like further details, or do you have any other questions?

Here are 5 questions to expand on this:

1. How could we change the payout values to make the game fair?
2. What is the expected value for North in this game?
3. How would adding jokers or wild cards affect the outcome?
4. What is the variance of this game's payouts?
5. How would the game change if the deck was smaller, say 32 cards?

Tip: In probability games, the expected value helps to determine whether a game is favorable or not in the long run.