A card is drawn from a standard​ 52-card deck. If the card is a​ king, you win ​$20​; ​otherwise, you lose ​$5. 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.
​E(X)equals

enter your response here dollars ​(Type an integer or a decimal rounded to the nearest cent as​ needed.)

To determine the expected value $E(X)$ of the game, we need to consider the probability of each outcome and the corresponding monetary value.

1. **Probability of drawing a king**:
   - There are 4 kings in a 52-card deck.
   - Probability of drawing a king: $P(\text{King}) = \frac{4}{52} = \frac{1}{13}$.

2. **Probability of not drawing a king**:
   - There are 52 - 4 = 48 non-king cards.
   - Probability of not drawing a king: $P(\text{Not King}) = \frac{48}{52} = \frac{12}{13}$.

3. **Monetary outcomes**:
   - If a king is drawn, you win $20.
   - If a king is not drawn, you lose $5.

4. **Expected value calculation**:
   - The expected value $E(X)$ is given by:
     \[
     E(X) = ( \text{Value if King} \times P(\text{King}) ) + ( \text{Value if Not King} \times P(\text{Not King}) )
     \]
   - Substituting the values:
     \[
     E(X) = (20 \times \frac{1}{13}) + (-5 \times \frac{12}{13})
     \]

5. **Calculations**:
   - Calculate each term:
     \[
     20 \times \frac{1}{13} = \frac{20}{13} \approx 1.54
     \]
     \[
     -5 \times \frac{12}{13} = -\frac{60}{13} \approx -4.62
     \]

   - Add the two terms together:
     \[
     E(X) = 1.54 + (-4.62) = 1.54 - 4.62 = -3.08
     \]

Therefore, the expected value of the game is $-\$3.08$.

If you have any questions or need further details, feel free to ask. Here are some additional questions you might be interested in:

1. How is the probability of drawing a specific card from a deck calculated?
2. What is the significance of the expected value in probability theory?
3. Can the expected value be positive in this type of game?
4. How would the expected value change if the winning amount was increased?
5. What are some real-life applications of expected value calculations?

**Tip:** Understanding expected value is crucial in making informed decisions in games of chance and various fields like finance and insurance.

Understand how to calculate the expected value of a game where drawing a king wins $20 and not drawing a king loses $5. Learn the step-by-step calculation involving probability theory and expected value formulas.

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