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We are given the probability distribution of a random variable \( X \), which represents the number of hits a baseball player obtains in the next 3 at-bats. The probabilities are:

- \( P(X = 1) = 0.2 \)
- \( P(X = 2) = 0.3 \)
- \( P(X = 0) = 2P(X = 3) \)

We are tasked with finding the expected value \( E(X) \).

### Step-by-step solution:

1. **Express \( P(X = 3) \) using the relationship given:**
   \[
   P(X = 0) = 2P(X = 3)
   \]
   Let \( P(X = 3) = p_3 \). Then:
   \[
   P(X = 0) = 2p_3
   \]

2. **The sum of all probabilities must equal 1:**
   \[
   P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 1
   \]
   Substituting the known probabilities:
   \[
   2p_3 + 0.2 + 0.3 + p_3 = 1
   \]
   Simplifying:
   \[
   3p_3 + 0.5 = 1
   \]
   \[
   3p_3 = 0.5
   \]
   \[
   p_3 = \frac{0.5}{3} = \frac{1}{6} \approx 0.1667
   \]
   Therefore, \( P(X = 3) = \frac{1}{6} \) and \( P(X = 0) = 2 \times \frac{1}{6} = \frac{2}{6} = \frac{1}{3} \).

3. **Now we calculate \( E(X) \):**
   The expected value \( E(X) \) is given by the formula:
   \[
   E(X) = 0 \times P(X = 0) + 1 \times P(X = 1) + 2 \times P(X = 2) + 3 \times P(X = 3)
   \]
   Substituting the values:
   \[
   E(X) = 0 \times \frac{1}{3} + 1 \times 0.2 + 2 \times 0.3 + 3 \times \frac{1}{6}
   \]
   Simplifying:
   \[
   E(X) = 0 + 0.2 + 0.6 + 0.5 = 1.3
   \]

Thus, the value of \( E(X) \) is \( \boxed{1.3} \), which corresponds to option (c).

Would you like further clarification on any steps?

### Related Questions:
1. How would the expected value change if \( P(X = 1) \) were increased?
2. Can the sum of probabilities ever exceed 1?
3. What is the variance for this probability distribution?
4. How does the relationship between \( P(X = 0) \) and \( P(X = 3) \) affect the solution?
5. What would the expected value be if \( X \) followed a uniform distribution?

### Tip:
Always remember that the sum of all probabilities in a probability distribution must equal 1, which is useful when solving for unknown probabilities.

Suppose that the random variable X is equal to the number of hits obtained by a certain baseball player in his next 3 at-bats. If P(X = 1) = 0.2, P(X = 2) = 0.3, and P(X = 0) = 2P(X = 3), then the value of E(X) is equal to?

Learn how to calculate the expected value (E(X)) for a baseball player's number of hits in 3 at-bats using a probability distribution. This problem involves key concepts such as random variables, expected value, and the law of total probability. Suitable for students in Grades 11-12.

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