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Let’s break down and solve this step-by-step.

### Problem Overview:
The hockey player takes 3 shots, and the probabilities of scoring on each shot **depend on previous outcomes**:
- **Initial probability**: 0.4 for the first shot.
- If he scores on the **first shot**, the second shot probability **increases by 0.1**.
- If he **misses the first**, the second shot probability **decreases by 0.1**.
- Similar rules apply to the third shot: the probability **increases by 0.1** if the player scores on the second shot and **decreases by 0.1** if he misses.

We need to complete:
- (a) **Probability distribution** of the total goals \(X\).
- (b) **Expected value** \(E(X)\).
- (c) **Standard deviation** \(SD(X)\).

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### Step 1: Analyze All Possible Sequences of Shots

For 3 shots, each can either be a goal (G) or miss (M). We need to:
- Find the probability of each sequence (e.g., GGG, GGM, etc.).
- Count the number of goals for each sequence to use in the distribution.

#### Possible Sequences:
There are \(2^3 = 8\) possible outcomes for the shots. Let’s list these and compute their probabilities.

1. **GGG** (3 goals):  
   - First: 0.4  
   - Second: 0.5 (because he scored the first)  
   - Third: 0.6 (because he scored the second)  
   - **Probability** = \(0.4 \times 0.5 \times 0.6 = 0.120\)

2. **GGM** (2 goals):  
   - First: 0.4  
   - Second: 0.5 (scored the first)  
   - Third: 0.4 (missed the second)  
   - **Probability** = \(0.4 \times 0.5 \times 0.4 = 0.080\)

3. **GMG** (2 goals):  
   - First: 0.4  
   - Second: 0.4 (missed the first)  
   - Third: 0.5 (scored the second)  
   - **Probability** = \(0.4 \times 0.4 \times 0.5 = 0.080\)

4. **GMM** (1 goal):  
   - First: 0.4  
   - Second: 0.4 (missed the first)  
   - Third: 0.3 (missed the second)  
   - **Probability** = \(0.4 \times 0.4 \times 0.3 = 0.048\)

5. **MGG** (2 goals):  
   - First: 0.6 (missed the first)  
   - Second:...

A hockey player takes 3 shots on a goalie. The probabilities of scoring on each shot depend on previous outcomes. The probability of scoring on the first shot is 0.4. If he scores, the second shot probability increases by 0.1; if he misses, it decreases by 0.1. The same pattern holds for the third shot, depending on the second shot's outcome.

This problem involves calculating the probability distribution, expected value, and standard deviation for the number of goals a hockey player scores based on changing probabilities after each shot. It demonstrates key probability concepts and statistical measures.

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