When the economic situation is "high", a certain economic indicator rises with probability
0.7. When the economic situation is "medium", the economic indicator rises with probability
0.4. When the economic situation is "low", the economic indicator rises with probability 0.2.
The economy is "high" 30% of the time, it is "medium" 50% of the time, and it is "low" 20%
of the time. Given that the indicator has just gone up, what is the probability that the
economic situation is high?

To solve this problem, we'll use **Bayes' theorem**. The problem asks for the probability that the economic situation is "high" given that the indicator has just gone up. 

Bayes' theorem is given by:

\[
P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}
\]

Here:

- \( A \) is the event that the economic situation is "high".
- \( B \) is the event that the indicator has gone up.

We are asked to find \( P(\text{High} | \text{Up}) \), which corresponds to \( P(A|B) \).

### Step 1: Determine the known probabilities

- \( P(\text{High}) = 0.3 \)
- \( P(\text{Medium}) = 0.5 \)
- \( P(\text{Low}) = 0.2 \)
- \( P(\text{Up}|\text{High}) = 0.7 \)
- \( P(\text{Up}|\text{Medium}) = 0.4 \)
- \( P(\text{Up}|\text{Low}) = 0.2 \)

### Step 2: Calculate \( P(\text{Up}) \)

Using the law of total probability:

\[
P(\text{Up}) = P(\text{Up}|\text{High}) \cdot P(\text{High}) + P(\text{Up}|\text{Medium}) \cdot P(\text{Medium}) + P(\text{Up}|\text{Low}) \cdot P(\text{Low})
\]

Substitute the known values:

\[
P(\text{Up}) = (0.7 \times 0.3) + (0.4 \times 0.5) + (0.2 \times 0.2)
\]

\[
P(\text{Up}) = 0.21 + 0.2 + 0.04 = 0.45
\]

### Step 3: Apply Bayes' Theorem

Now, using Bayes' theorem:

\[
P(\text{High}|\text{Up}) = \frac{P(\text{Up}|\text{High}) \cdot P(\text{High})}{P(\text{Up})}
\]

Substitute the values:

\[
P(\text{High}|\text{Up}) = \frac{0.7 \times 0.3}{0.45} = \frac{0.21}{0.45} \approx 0.467
\]

### Final Answer:

The probability that the economic situation is "high" given that the indicator has gone up is approximately **0.467** or **46.7%**.

Would you like more details on this solution or have any questions?

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Here are 5 related questions you might find interesting:

1. How does Bayes' theorem apply to diagnostic tests in medicine?
2. What is the law of total probability and how is it used in conditional probability?
3. If the economic situation probabilities were different, how would that affect the final result?
4. Can Bayes' theorem be used to update probabilities in real-time as new information is received?
5. What are some real-world applications of Bayes' theorem in finance?

**Tip:** In problems involving conditional probability, identifying the correct prior and likelihoods is crucial for applying Bayes' theorem correctly.

Explore how Bayes' theorem is used to determine the probability that the economic situation is high given that the economic indicator has just gone up. Includes step-by-step calculations and explanations of conditional probability and the law of total probability.

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