To calculate the expected value (EV) of this proposition, we need to consider both the probability of drawing a card with a value of four or less and the associated outcomes.

1. Identify the number of cards with values of four or less:

• A standard deck contains 52 cards.
• Cards with a value of four or less include 2, 3, and 4.
• Each of these values has four cards (one from each suit), for a total of:
  • $3 \text{ values} \times 4 \text{ cards per value} = 12 \text{ cards with values 4 or less}$

2. Find the probability of drawing a card with a value of four or less:

$\text{Probability} = \frac{12}{52} = \frac{3}{13} \approx 0.2308$

3. Find the probability of drawing any other card:

There are 52 cards in total, and 12 of them are 4 or less, so: $\text{Probability of other cards} = 1 - \frac{12}{52} = \frac{40}{52} = \frac{10}{13} \approx 0.7692$

4. Calculate the expected value (EV):

Now, we calculate the expected value using the formula: $\text{EV} = (\text{Probability of winning} \times \text{Payoff from winning}) + (\text{Probability of losing} \times \text{Payoff from losing})$

• The payoff for winning (drawing a card of 4 or less) is $134.
• The payoff for losing (drawing any other card) is -$37 (since you pay this amount).

$\text{EV} = \left( \frac{12}{52} \times 134 \right) + \left( \frac{40}{52} \times (-37) \right)$ $\text{EV} = \left( \frac{3}{13} \times 134 \right) + \left( \frac{10}{13} \times (-37) \right)$

5. Perform the calculations:

$\text{EV} = \left( 0.2308 \times 134 \right) + \left( 0.7692 \times (-37) \right)$

$\text{EV} = 30.91 + (-28.46) = 2.45$

Thus, the expected value of the proposition is approximately $2.45.

Conclusion:

The expected value is positive at $2.45, meaning on average, you would expect to win $2.45 each time this bet is played.

Would you like more details or have any questions?

Here are 5 related questions:

1. How is expected value applied in real-life gambling scenarios?
2. What happens if Aces are considered the lowest card instead of the highest?
3. How does increasing the winning payout affect the expected value?
4. What if there were only three suits in the deck—how would the probabilities change?
5. How can expected value help in making investment decisions?

Tip: Always consider both the probability and the outcomes (payoffs) when calculating the expected value. This helps in evaluating whether a gamble or decision is favorable!