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.