If the probability of a random variable X is a given below :
`{:(X=x:,-2,-1,0,1,2,3),(P(X=x):,(1)/(10),k,(1)/(5),2k,(3)/(10),k):}`
Then the value of k, is :

Options

A. `(1)/(10)`
B. `(2)/(10)`
C. `(3)/(10)`
D. `(7)/(10)`

Explanation

To find the value of \( k \) in the given probability distribution, we need to ensure that the sum of all probabilities equals 1. Let's break down the solution step by step.

### Step 1: Write down the probabilities
The given probabilities for the random variable \( X \) are:
- \( P(X = -2) = \frac{1}{10} \)
- \( P(X = -1) = k \)
- \( P(X = 0) = \frac{1}{5} \)
- \( P(X = 1) = 2k \)
- \( P(X = 2) = \frac{3}{10} \)
- \( P(X = 3) = k \)

### Step 2: Set up the equation
According to the property of probability distributions, the sum of all probabilities must equal 1:
\[ P(X = -2) + P(X = -1) + P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) = 1 \]
Substituting the values, we get:
\[ \frac{1}{10} + k + \frac{1}{5} + 2k + \frac{3}{10} + k = 1 \]

### Step 3: Simplify the equation
First, convert \( \frac{1}{5} \) to a fraction with a denominator of 10:
\[ \frac{1}{5} = \frac{2}{10} \]
Now substitute this back into the equation:
\[ \frac{1}{10} + k + \frac{2}{10} + 2k + \frac{3}{10} + k = 1 \]
Combine like terms:
\[ \frac{1 + 2 + 3}{10} + (k + 2k + k) = 1 \]
This simplifies to:
\[ \frac{6}{10} + 4k = 1 \]

### Step 4: Solve for \( k \)
Now, isolate \( k \):
\[ 4k = 1 - \frac{6}{10} \]
Convert 1 to a fraction with a denominator of 10:
\[ 1 = \frac{10}{10} \]
Thus,
\[ 4k = \frac{10}{10} - \frac{6}{10} = \frac{4}{10} \]
Now, divide both sides by 4:
\[ k = \frac{4}{10} \div 4 = \frac{4}{40} = \frac{1}{10} \]

### Conclusion
The value of \( k \) is \( \frac{1}{10} \).

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