Billy is deciding whether or not to have surgery. If he has the surgery, there is a 0.9 chance that he will recover full health and 0.1 chance that the surgery will have no effect on his condition. For Billy, having the surgery and thereby recovering full health has utility 100. Moreover, for Billy, having the surgery and this having no effect on his condition has utility -50.
For Billy, what is the expected utility of having the surgery?
(Please answer with a number rounded to three decimal places rather than with a fraction, e.g. 0.111 rather than 1/9).

To find the expected utility of having the surgery for Billy, we'll use the probabilities and utilities of each possible outcome.

1. Probability of recovering full health: \( P(\text{full health}) = 0.9 \)
2. Utility of recovering full health: \( U(\text{full health}) = 100 \)

3. Probability of no effect: \( P(\text{no effect}) = 0.1 \)
4. Utility of no effect: \( U(\text{no effect}) = -50 \)

The expected utility \( E(U) \) is calculated as follows:

\[
E(U) = P(\text{full health}) \times U(\text{full health}) + P(\text{no effect}) \times U(\text{no effect})
\]

Substituting the values:

\[
E(U) = (0.9 \times 100) + (0.1 \times -50)
\]

Performing the calculations:

\[
E(U) = 90 + (-5) = 90 - 5 = 85
\]

So, the expected utility of having the surgery for Billy is:

\[
85.000
\]

Would you like more details or have any questions? Here are some questions you might consider next:

1. What are the implications of expected utility in decision making?
2. How would the expected utility change if the probabilities were different?
3. Can you explain the concept of utility in more depth?
4. How does expected utility theory apply in real-world scenarios?
5. What are some limitations of using expected utility in decision making?

**Tip:** When dealing with probabilities and utilities, ensure that all probabilities sum to 1 and carefully consider each outcome's impact.

Understand how to calculate the expected utility of a surgical decision for Billy. This problem involves probabilities of recovery and no effect, along with corresponding utilities. Learn the application of expected utility theory in decision-making scenarios.

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