A card is drawn from a standard​ 52-card deck. If the card is a​ king, you win ​$20. ​Otherwise, you lose ​$1. What is the expected value of the​ game?
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Part 1
Let X be the random variable for the amount won on a single play of this game. Fill in the probabilities for the probability distribution of the random variable X.
Start 2 By 3 Table 1st Row 1st Column x Subscript i 2nd Column $ 20 3rd Column negative $ 1 2nd Row 1st Column p Subscript i 2nd Column nothing 3rd Column nothing EndTable

​(Type integers or simplified​ fractions.)

To find the expected value of the game, we first need to fill in the probability distribution of the random variable $ X $, where $ X $ represents the amount won on a single play of the game.

### Step 1: Determine the probabilities for each outcome.

- **Probability of winning $20 (drawing a King):**
  
  There are 4 Kings in a standard 52-card deck, so the probability of drawing a King is:
  \[
  P(\text{King}) = \frac{4}{52} = \frac{1}{13}
  \]

- **Probability of losing $1 (drawing any card other than a King):**
  
  There are 48 other cards (non-Kings) in the deck, so the probability of drawing one of these is:
  \[
  P(\text{Non-King}) = \frac{48}{52} = \frac{12}{13}
  \]

### Step 2: Fill in the probability distribution table.

| $ x_i $   | $20  | $-$1) |
|-------------|------|-------|
| \( p_i $   | $\frac{1}{13}$ | $\frac{12}{13}$ |

### Step 3: Calculate the expected value.

The expected value $ E(X) $ is calculated using the formula:
\[
E(X) = \sum_{i} x_i \cdot p_i
\]

Substitute the values:
\[
E(X) = 20 \times \frac{1}{13} + (-1) \times \frac{12}{13}
\]

Simplify the expression:
\[
E(X) = \frac{20}{13} - \frac{12}{13} = \frac{8}{13}
\]

Thus, the expected value of the game is $ \frac{8}{13} $, which is approximately $0.62.

### Conclusion
The expected value of the game is approximately **$0.62**.

Would you like more details or have any questions?

### Relative Questions:
1. What is the probability of drawing a King in two consecutive draws?
2. How would the expected value change if you won $25 instead of $20 for drawing a King?
3. What is the expected value if there are two cards drawn with replacement?
4. How would the game change if there were Jokers in the deck?
5. What is the variance and standard deviation of the game?

### Tip:
In probability, the expected value gives a measure of the center of the distribution, often representing a long-term average outcome of a random process.

Learn how to calculate the expected value of a game involving drawing cards from a deck. Understand the probabilities of drawing a King or a non-King, and apply the expected value formula. Suitable for advanced high school students and anyone interested in probability theory.

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