Question: User: top-center: table with columns $s_1$ and $s_2$; row A has 10 under $s_1$ and 0 under $s_2$; row B has 9 under $s_1$ and 4 under $s_2$.
For what probability $P(s_1)$ are the acts equally preferred under expected utility?
0.1
0.5
0.6
0.8
None of the above.consider the displayed decision matrix
1. **State the problem:** We have two acts, A and B, with utilities given for two states $s_1$ and $s_2$.
The utilities are:
- Act A: $u(A,s_1) = 10$, $u(A,s_2) = 0$
- Act B: $u(B,s_1) = 9$, $u(B,s_2) = 4$
We want to find the probability $P(s_1)$ such that the expected utilities of acts A and B are equal.
2. **Formula for expected utility:**
$$EU(\text{act}) = P(s_1) \times u(\text{act}, s_1) + (1 - P(s_1)) \times u(\text{act}, s_2)$$
3. **Set expected utilities equal:**
$$P(s_1) \times 10 + (1 - P(s_1)) \times 0 = P(s_1) \times 9 + (1 - P(s_1)) \times 4$$
4. **Simplify the equation:**
$$10P(s_1) = 9P(s_1) + 4 - 4P(s_1)$$
$$10P(s_1) = 9P(s_1) + 4 - 4P(s_1)$$
$$10P(s_1) = (9P(s_1) - 4P(s_1)) + 4$$
$$10P(s_1) = 5P(s_1) + 4$$
5. **Isolate $P(s_1)$:**
$$10P(s_1) - 5P(s_1) = 4$$
$$\cancel{10}P(s_1) - \cancel{5}P(s_1) = 4$$
$$5P(s_1) = 4$$
$$P(s_1) = \frac{4}{5} = 0.8$$
6. **Interpretation:** The acts are equally preferred when the probability of state $s_1$ is $0.8$.
**Final answer:** $P(s_1) = 0.8$