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To complete the probability distribution for the random variable \( X \), we need to ensure two conditions for the probabilities:

1. The sum of all probabilities must equal 1.
2. Each probability must be non-negative.

Given the table:

| Value \( x \) of \( X \) | \( P(X = x) \)  |
|------------------------|------------------|
| -2                     | 0.10             |
|  2                     | 0.18             |
|  3                     | ?                |
|  4                     | 0.11             |
|  6                     | ?                |

### Step 1: Set up the equation for the total sum of probabilities.
The total must sum to 1, so:

\[
0.10 + 0.18 + P(X=3) + 0.11 + P(X=6) = 1
\]

### Step 2: Solve for the sum of \( P(X = 3) + P(X = 6) \).
\[
0.10 + 0.18 + 0.11 = 0.39
\]
\[
P(X = 3) + P(X = 6) = 1 - 0.39 = 0.61
\]

### Step 3: Distribute the remaining probability.
This leaves \( P(X = 3) + P(X = 6) = 0.61 \). Without further constraints, one possible solution is to assign equal probabilities:

\[
P(X = 3) = 0.305 \quad \text{and} \quad P(X = 6) = 0.305
\]

This would create a valid probability distribution.

Would you like more details on how to adjust this or have any specific questions?

Here are 5 follow-up questions related to probability distributions:

1. What is a probability distribution, and how is it defined for discrete variables?
2. How do you verify if a given set of probabilities forms a valid distribution?
3. How can you find the expected value (mean) of a discrete random variable?
4. What is the significance of the variance and standard deviation in probability distributions?
5. How can probability distributions be applied in real-world scenarios, such as decision-making?

**Tip**: When solving probability questions, always ensure the total sum of probabilities equals 1, and each individual probability must be between 0 and 1.

Fill in the P(X = x) values to give a legitimate probability distribution for the discrete random variable X, whose possible values are −2, 2, 3, 4, and 6.

Learn how to complete a probability distribution for a discrete random variable with values −2, 2, 3, 4, and 6. Ensure the sum of probabilities equals 1 with step-by-step calculations suitable for Grades 9-12 students.

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