To find the expected value of the score for a randomly chosen card hand, we need to understand the scoring system. Let's denote:

• $K$ as the number of kings in the hand.
• $C$ as the number of clubs in the hand.

The score $S$ is given by:

$S = 2K - C$

1. Number of Kings (K)

There are 4 kings in a standard deck of 52 cards. The possible values for $K$ range from 0 to 4.

2. Number of Clubs (C)

There are 13 clubs in a deck. The possible values for $C$ range from 0 to 13.

3. Probability of Drawing a King or Club

Each card is equally likely to be drawn, and the presence of one card does not affect the others in the case of large enough hands (as long as not all cards are drawn). For simplicity and assuming the hand size allows it, the probability of drawing a king or a club independently remains 4/52 for a king and 13/52 for a club.

4. Expected Value Calculation

To find the expected value $E[S]$, we calculate:

[ E[S] = E[2K - C] = 2E[K] - E[C] ]

Where $E[K]$ and $E[C]$ are the expected number of kings and clubs in a randomly chosen hand, respectively.

• Expected number of kings, $E[K]$: Since there are 4 kings in a deck of 52 cards, the probability of drawing a king is $\frac{4}{52} = \frac{1}{13}$.

  Thus, the expected number of kings in a randomly chosen card is: $E[K] = 1 \cdot \frac{1}{13} + 0 \cdot \frac{12}{13} = \frac{1}{13}$

• Expected number of clubs, $E[C]$: Since there are 13 clubs in a deck of 52 cards, the probability of drawing a club is $\frac{13}{52} = \frac{1}{4}$.

  Thus, the expected number of clubs in a randomly chosen card is: $E[C] = 1 \cdot \frac{1}{4} + 0 \cdot \frac{3}{4} = \frac{1}{4}$

Plugging these values into our equation for $E[S]$:

$E[S] = 2 \cdot \frac{1}{13} - \frac{1}{4} = \frac{2}{13} - \frac{1}{4}$

To simplify:

$E[S] = \frac{2}{13} - \frac{13}{52} = \frac{2}{13} - \frac{13}{52} = \frac{8 - 13}{52} = \frac{-5}{52} = -\frac{5}{52}$

Thus, the expected value of the score is $-\frac{5}{52}$.

Would you like a detailed breakdown of any part of this solution? Here are some questions to consider next:

1. How are probabilities calculated for different card types in a deck?
2. What is the concept of expected value in probability?
3. How does the independence of events affect probability calculations?
4. Can you explain how to find the expected value for different hands sizes?
5. What are the implications of changing the scoring system?

Tip: Always double-check calculations for accuracy, especially when dealing with fractions and probability.