d and e are 2 units. d = 1 and e = 1/16d. The probability of d being 16e is 0.6 but can go up to 32e with a probability of 0.4. If D Gives me 12% intrest and e 24 which one is better if i invest 6000d in each d and e. Convert accordingly

Let's break down the problem step by step.

### Given Information:
1. \( d = 1 \) and \( e = \frac{1}{16}d = \frac{1}{16} \).
2. \( d \) can either be \( 16e \) with a probability of 0.6 or \( 32e \) with a probability of 0.4.
3. Interest rates:
   - Investment in \( d \) gives 12% interest.
   - Investment in \( e \) gives 24% interest.
4. Investment amount: \( 6000d \) in both \( d \) and \( e \).

### Step 1: Understand \( d \) in terms of \( e \)
- Given \( d = 16e \) with a probability of 0.6.
- Given \( d = 32e \) with a probability of 0.4.

### Step 2: Expected Value of \( d \) 
Calculate the expected value \( E(d) \):
\[
E(d) = 0.6 \times 16e + 0.4 \times 32e = 9.6e + 12.8e = 22.4e
\]

### Step 3: Convert \( d \) to \( e \) 
Given \( d = 1 \) and \( e = \frac{1}{16} \):

- For \( 6000d \) investment:
  \[
  6000d = 6000 \times 1 = 6000 \text{ units of } d
  \]
  Convert this to \( e \):
  \[
  \text{Investment in terms of } e: 6000d = 6000 \times 16e = 96000e
  \]

- For \( 6000e \) investment:
  \[
  6000e = 6000 \times \frac{1}{16}d = 375d
  \]

### Step 4: Calculate Returns
1. **Return from \( d \) investment:**
   - Interest: \( 12\% \) on 6000 units of \( d \).
   \[
   \text{Return} = 6000 \times 1 \times 0.12 = 720 \text{ units of } d
   \]
   Convert 720 units of \( d \) to \( e \):
   \[
   720d = 720 \times 16e = 11520e
   \]

2. **Return from \( e \) investment:**
   - Interest: \( 24\% \) on 96000 units of \( e \).
   \[
   \text{Return} = 96000 \times 0.24 = 23040e
   \]

### Conclusion:
- **Investment in \( d \)** yields a return equivalent to \( 11520e \).
- **Investment in \( e \)** yields a return of \( 23040e \).

### Which is better?
- **Investing in \( e \)** is better since it yields a return of 23040 units of \( e \) compared to 11520 units of \( e \) when investing in \( d \).

### Questions:
1. How would the outcome change if the interest rates were different for \( d \) and \( e \)?
2. What happens if the probabilities for \( d = 16e \) and \( d = 32e \) change?
3. How would the result be different if the initial investments were in terms of \( e \) instead of \( d \)?
4. What if you invested different amounts in \( d \) and \( e \) instead of equal amounts?
5. How does the concept of expected value influence investment decisions in this scenario?

**Tip:** Always convert all variables to a common unit to make comparisons easier when dealing with different scales or probabilities.

Explore the comparison of investments between d and e given different interest rates and probabilities. Calculate expected values and returns for investments of 6000 units in d and e, considering 12% interest for d and 24% interest for e.

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