	s1	s2
d1	200	400
d2	500	300
 
P(F) = 0.62
P(U) = 0.38
P(s1|F) = 0.64
P(s2|F) = 0.36
P(s1|U) = 0.14
P(s2|U) = 0.86
P(s1) = 0.44
P(s2) = 0.56

Make a decision tree. What is the EV of node 3?

To create a decision tree and calculate the Expected Value (EV) of node 3, let's break down the problem step by step.

### Step 1: Understanding the Problem

We are given:
- Probabilities associated with two events \( F \) and \( U \) (P(F) and P(U)).
- Conditional probabilities for \( s1 \) and \( s2 \) given \( F \) and \( U \) (P(s1|F), P(s2|F), P(s1|U), and P(s2|U)).
- The overall probabilities of \( s1 \) and \( s2 \) (P(s1) and P(s2)).

The decision tree will consist of nodes that split based on these probabilities.

### Step 2: Constructing the Decision Tree

1. **Initial Decision**: Split based on whether the event is \( F \) or \( U \).
   - Probability of \( F \): \( P(F) = 0.62 \)
   - Probability of \( U \): \( P(U) = 0.38 \)

2. **Next Split (Given \( F \) or \( U \))**:
   - Given \( F \):
     - Probability of \( s1 \) given \( F \): \( P(s1|F) = 0.64 \)
     - Probability of \( s2 \) given \( F \): \( P(s2|F) = 0.36 \)
   - Given \( U \):
     - Probability of \( s1 \) given \( U \): \( P(s1|U) = 0.14 \)
     - Probability of \( s2 \) given \( U \): \( P(s2|U) = 0.86 \)

### Step 3: Decision Tree Layout

Here's a textual representation of the decision tree:

```
          Start
          /   \
       F       U
     /   \    /   \
  s1     s2 s1     s2
 (N1)  (N2) (N3)  (N4)
```

- Nodes \( N1 \) and \( N2 \) follow from \( F \).
- Nodes \( N3 \) and \( N4 \) follow from \( U \).

### Step 4: Calculate the Expected Value of Node 3

Node 3 corresponds to \( s1 \) given \( U \).

- **Value of Node 3**: The value associated with \( s1 \) under \( U \) is given in the table: 500.
- **Probability of reaching Node 3**: 
  \[
  P(\text{Node 3}) = P(U) \times P(s1|U) = 0.38 \times 0.14 = 0.0532
  \]

- **Expected Value at Node 3**:
  \[
  \text{EV of Node 3} = P(\text{Node 3}) \times \text{Value at Node 3} = 0.0532 \times 500 = 26.6
  \]

### Conclusion

The Expected Value (EV) of Node 3 is **26.6**.

Would you like more details, or do you have any questions about this calculation?

Here are some related questions:

1. What is the Expected Value of Node 1?
2. What is the Expected Value of Node 4?
3. How would the decision tree change if \( P(F) \) and \( P(U) \) were different?
4. What is the total Expected Value of the decision tree?
5. How do the conditional probabilities affect the decision-making process?

**Tip**: Always verify the probabilities used in each branch of the decision tree to ensure the accuracy of the Expected Value calculations.

Make a decision tree. What is the EV of node 3?

This guide walks through the process of constructing a decision tree based on given probabilities and calculating the Expected Value (EV) of a specific node, focusing on how to find the EV of Node 3 in the context of probabilities related to events F and U. Ideal for students in Grades 11-12.

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