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**Probabilistic Dynamic Programming**:  
You wish to invest $3 million over the next 5 years. There is a 60% probability that the money will double and a 40% probability of losing the entire investment. Indicate the probable accumulated amount at the end of the fifth year.

To solve this, we calculate the expected value of the investment at the end of 5 years, taking into account the probabilities of each outcome.

**Expected Value Calculation**:  
1. **Doubling the Investment**: The investment grows to $6 million with a probability of 60%.
2. **Losing the Investment**: The investment goes to $0 with a probability of 40%.

The expected value $ EV $ of the investment at the end of 5 years can be calculated using the formula:
\[ EV = (Probability \; of \; Doubling) \times (Value \; if \; Doubled) + (Probability \; of \; Losing) \times (Value \; if \; Lost) \]

Let's calculate this:

\[ EV = (0.60 \times 6 \, \text{million}) + (0.40 \times 0) \]

Executing this calculation gives:The probable accumulated amount at the end of the fifth year is approximately $3.6 million.

Would you like more details on this calculation or have any other questions? Here are some related topics you might find interesting:

1. How would the expected value change if the investment period is extended beyond 5 years?
2. What is the impact of changing the probabilities of doubling or losing the investment on the expected value?
3. How would introducing a third possible outcome, such as a partial return of the investment, affect the calculations?
4. Can we model the probability changes over each year instead of having them constant for the entire 5 years?
5. How do real-world factors like inflation or changing economic conditions affect such investment calculations?

**Tip:** When considering investments, always factor in the risk versus return trade-offs to align with your financial goals and risk tolerance.

Se desea invertir $3 millones durante los 5 años venideros. Se estima en 60% la probabilidad de que el dinero aumente al doble y 40% la probabilidad de perder la cantidad invertida. Indique el monto probable acumulado al quinto año.

Learn how to calculate the expected value for a $3 million investment over 5 years with a 60% chance of doubling and a 40% chance of losing the investment. Explore key probability concepts and expected value calculations.

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