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The image contains a question regarding two investments (Investment A and Investment B) and a utility function $ U(W) = -W - 0.04W^2 $. Let's break down the steps for solving both questions.

### 1. Compare the two investments using the given utility function.

#### Investment A
Investment A has the following outcomes and probabilities:
- $10 with probability 0.4
- $14 with probability 0.4
- $7 with probability 0.2

We calculate the expected utility by applying the utility function $ U(W) = -W - 0.04W^2 $ for each outcome.

\[
U(10) = -10 - 0.04(10^2) = -10 - 0.04(100) = -10 - 4 = -14
\]
\[
U(14) = -14 - 0.04(14^2) = -14 - 0.04(196) = -14 - 7.84 = -21.84
\]
\[
U(7) = -7 - 0.04(7^2) = -7 - 0.04(49) = -7 - 1.96 = -8.96
\]

Now, calculate the expected utility for Investment A by multiplying the utility of each outcome by its probability:
\[
E[U(A)] = 0.4 \times (-14) + 0.4 \times (-21.84) + 0.2 \times (-8.96)
\]
\[
E[U(A)] = -5.6 - 8.736 - 1.792 = -16.128
\]

#### Investment B
Investment B has the following outcomes and probabilities:
- $5 with probability 0.5
- $12 with probability 0.25
- $20 with probability 0.25

Let's compute the utility for each outcome in Investment B:
\[
U(5) = -5 - 0.04(5^2) = -5 - 0.04(25) = -5 - 1 = -6
\]
\[
U(12) = -12 - 0.04(12^2) = -12 - 0.04(144) = -12 - 5.76 = -17.76
\]
\[
U(20) = -20 - 0.04(20^2) = -20 - 0.04(400) = -20 - 16 = -36
\]

Now, calculate the expected utility for Investment B:
\[
E[U(B)] = 0.5 \times (-6) + 0.25 \times (-17.76) + 0.25 \times (-36)
\]
\[
E[U(B)] = -3 - 4.44 - 9 = -16.44
\]

### Which investment is preferred?
The expected utility of Investment A is -16.128, and the expected utility of Investment B is -16.44. Since we are dealing with negative utilities, **Investment A** is preferred because it has a higher (less negative) expected utility.

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### 2. Adjust probabilities for indifference between the two investments

The second part asks for the adjustment of probabilities such that the investor is indifferent between Investment A and Investment B. Indifference occurs when the expected utilities of both investments are equal.

We want to adjust the probability of the $5 payoff (currently 0.5) and the $12 payoff (currently 0.25) in Investment B to make $ E[U(A)] = E[U(B)] $.

Let $ p_5 $ be the new probability of the $5 payoff, and $ p_{12} $ be the new probability of the $12 payoff. We can set up the following equation:

\[
E[U(A)] = E[U(B)]
\]
\[
-16.128 = p_5 \times (-6) + p_{12} \times (-17.76) + (1 - p_5 - p_{12}) \times (-36)
\]

We can now solve this equation for $ p_5 $ and $ p_{12} $.

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Would you like a detailed step-by-step calculation for adjusting the probabilities, or should I proceed with simplifying the equation further?

Consider the following two investments. Which is preferred if the utility function is U(W) = -W - 0.04W^2? Also, how would the probabilities of the outcomes in Investment B have to change so that the investor is indifferent between Investments A and B?

This problem involves comparing two investments based on an expected utility calculation with the given utility function U(W) = -W - 0.04W^2. The solution includes step-by-step calculations of expected utility and adjustments of probabilities to achieve indifference between two investment choices.

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