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

To compare the returns from investing in \( D \) and \( e \), we need to calculate the expected returns based on the given interest rates, the initial investment of 6000 units, and the probability distribution of \( e \).

### Given:
- \( d = 1 \)
- \( e = \frac{1}{16}d = \frac{1}{16} \times 1 = \frac{1}{16} \)
- Probability of \( e \) being 16: 0.6
- Probability of \( e \) going up to 32: 0.4
- Interest rate for \( D \): 12% or 0.12
- Interest rate for \( e \): 24% or 0.24
- Total investment: 6000 units, distributed equally in both \( D \) and \( e \), so 3000 units in \( D \) and 3000 units in \( e \).

### Steps:

#### 1. **Return from Investment in \( D \):**
   - The investment in \( D \) is 3000 units.
   - Return from \( D \) at 12% interest:
     \[
     \text{Return from } D = 3000 \times 0.12 = 360 \text{ units}
     \]

#### 2. **Expected Return from Investment in \( e \):**
   - The investment in \( e \) is 3000 units.
   - The value of \( e \) can be either 16 or 32 with respective probabilities of 0.6 and 0.4.

   **Expected value of \( e \):**
   \[
   E(e) = (16 \times 0.6) + (32 \times 0.4) = 9.6 + 12.8 = 22.4
   \]

   **Expected Return from \( e \):**
   \[
   \text{Expected Return from } e = E(e) \times 0.24 \times 3000
   \]
   \[
   \text{Expected Return from } e = 22.4 \times 0.24 \times 3000 = 1612.8 \text{ units}
   \]

### 3. **Total Expected Returns:**
- Total return from \( D \) and \( e \):
  \[
  \text{Total Expected Return} = 360 + 1612.8 = 1972.8 \text{ units}
  \]

### Conclusion:
If you invest 6000 units, with 3000 in \( D \) and 3000 in \( e \), the total expected return would be 1972.8 units. The return from \( e \) is much higher (1612.8 units) compared to \( D \) (360 units), making \( e \) the better option in terms of expected returns.

Would you like more details or have any questions?

### Follow-up Questions:
1. How would the decision change if you could invest different amounts in \( D \) and \( e \)?
2. What happens if the interest rates for \( D \) and \( e \) change?
3. How does the risk associated with \( e \) compare to the guaranteed return from \( D \)?
4. What if you reinvest the

Learn how to compare investment returns between two options (D and e) using probability and interest rates. Calculate expected returns with a detailed solution example. Suitable for high school students studying probability and finance.

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