A card is drawn from a standard​ 52-card deck. If the card is a​ king, you win ​$40. ​Otherwise, you lose ​$3. 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 $ 40 3rd Column negative $ 3 2nd Row 1st Column p Subscript i 2nd Column nothing 3rd Column nothing EndTable

​(Type integers or simplified​ fractions.)

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

A standard 52-card deck has 4 kings.

- The probability of drawing a king (and winning $40) is:
  \[
  p(\text{king}) = \frac{4}{52} = \frac{1}{13}
  \]

- The probability of not drawing a king (and losing $3) is:
  \[
  p(\text{not king}) = 1 - p(\text{king}) = 1 - \frac{1}{13} = \frac{12}{13}
  \]

We can now fill in the probabilities for the probability distribution of the random variable $ X $:

\[
\begin{array}{c|c|c}
x_i & 40 & -3 \\
\hline
p_i & \frac{1}{13} & \frac{12}{13} \\
\end{array}
\]

The expected value $ E(X) $ of the game is calculated using the formula for the expected value of a discrete random variable:

\[
E(X) = \sum (x_i \cdot p_i)
\]

Substituting the values from our table:

\[
E(X) = 40 \cdot \frac{1}{13} + (-3) \cdot \frac{12}{13}
\]

\[
E(X) = \frac{40}{13} + \frac{-36}{13}
\]

\[
E(X) = \frac{40 - 36}{13}
\]

\[
E(X) = \frac{4}{13}
\]

Therefore, the expected value of the game is $\frac{4}{13}$ dollars, or approximately $0.31.

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. How do you calculate the variance and standard deviation of this game?
2. What is the probability of winning exactly once in two plays of the game?
3. If you play the game 52 times, what is the expected total winnings or losses?
4. How does the expected value change if the winning amount is increased to $50?
5. What is the probability of losing money after 10 plays of the game?
6. How do you determine the break-even point for the game in terms of number of plays?
7. Can you derive a general formula for the expected value if the winning amount and the losing amount are variables?
8. What would the expected value be if there were 6 kings in the deck instead of 4?

**Tip:** When calculating probabilities involving a deck of cards, always ensure to account for the total number of cards and specific counts for each type (like kings, aces, etc.).

Understand how to calculate the expected value of a game involving drawing cards from a standard 52-card deck. Learn the probability distribution, expected value formula, and step-by-step calculation for the game where drawing a king wins $40 and not drawing a king loses $3.

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