If you draw a card with a value of four or less from a standard deck of cards, I will pay you $41
.  If not, you pay me $16
.  (Aces are considered the highest card in the deck.)

Step 1 of 2 :  Find the expected value of the proposition. Round your answer to two decimal places. Losses must be expressed as negative values.

To calculate the expected value (EV) of the proposition, we'll break it down step by step.

### Step 1: Determine the probability of drawing a card with a value of four or less.
In a standard deck of 52 cards, there are 4 suits, and each suit contains the following cards with values four or less:
- 2, 3, 4 (no Aces since they are considered the highest card)

This means there are $ 3 $ cards (2, 3, 4) per suit, and there are 4 suits. So the total number of cards with a value of four or less is:
\[
3 \text{ cards per suit} \times 4 \text{ suits} = 12 \text{ cards with a value of four or less}.
\]

Therefore, the probability of drawing a card with a value of four or less is:
\[
P(\text{4 or less}) = \frac{12}{52} = \frac{3}{13}.
\]

### Step 2: Determine the probability of drawing a card with a value greater than four.
Since there are 52 cards in total and 12 cards have a value of four or less, there are $ 52 - 12 = 40 $ cards with a value greater than four.

The probability of drawing a card with a value greater than four is:
\[
P(\text{greater than 4}) = \frac{40}{52} = \frac{10}{13}.
\]

### Step 3: Calculate the expected value.
The expected value is calculated as:
\[
EV = (\text{P(win)} \times \text{payoff for win}) + (\text{P(loss)} \times \text{payoff for loss}).
\]

In this case:
- The payoff for winning (drawing a card with a value of 4 or less) is $41.
- The payoff for losing (drawing a card with a value greater than 4) is -$16 (a loss is expressed as a negative value).

So, the expected value is:
\[
EV = \left( \frac{3}{13} \times 41 \right) + \left( \frac{10}{13} \times (-16) \right).
\]

Let's calculate this.The expected value of the proposition is $-2.85$. This means, on average, you would expect to lose $2.85 per draw over time.

Would you like further details or have any other questions?

Here are 5 related questions you might explore:
1. How does changing the payout affect the expected value?
2. What happens to the expected value if aces are considered low?
3. How do probabilities change with a different deck size?
4. How can you calculate expected value for more complex games of chance?
5. What is the variance and standard deviation of this proposition?

**Tip:** When calculating expected values, always ensure you have accounted for all possible outcomes and their respective probabilities.

If you draw a card with a value of four or less from a standard deck of cards, I will pay you $41. If not, you pay me $16. (Aces are considered the highest card in the deck.) Find the expected value of the proposition. Round your answer to two decimal places. Losses must be expressed as negative values.

Calculate the expected value of drawing a card with a value of four or less from a standard deck of cards, considering a payout of $41 for winning and a loss of $16 for losing. This problem uses probability and expected value concepts, suitable for students in Grades 9-12.

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