The question from your image states the following:

"A certain game consists of rolling a single fair die and pays off as follows: $10 for a 6, $8 for a 5, $3 for a 4, and no payoff otherwise. Find the expected winnings for this game (Round to the nearest hundredth)."

To calculate the expected winnings:

1. A fair die has six outcomes, each with a probability of $\frac{1}{6}$.
2. The outcomes and their corresponding payoffs are:
   • Rolling a 6: $10
   • Rolling a 5: $8
   • Rolling a 4: $3
   • Rolling a 1, 2, or 3: $0 (no payoff)

Expected Value Formula:

$E(X) = \sum ( \text{Payoff} \times \text{Probability})$

The expected value for each outcome can be calculated as: $E(X) = \left( \frac{1}{6} \times 10 \right) + \left( \frac{1}{6} \times 8 \right) + \left( \frac{1}{6} \times 3 \right) + \left( \frac{3}{6} \times 0 \right)$

Now I'll calculate the expected winnings.The expected winnings for this game are $3.50, rounded to the nearest hundredth.

Would you like more details on how this was calculated or have any other questions?

Here are some related questions to consider:

1. What is the concept of expected value and how is it used in probability?
2. How would the expected value change if the payoffs were different?
3. How is the probability of each outcome derived for a fair die?
4. Can the expected value ever be negative in such games? Under what conditions?
5. How would the result differ if the die was biased (i.e., not fair)?

Tip: In games of chance, expected value helps determine the long-term average outcome, which is useful for decision-making and risk analysis.