The probability distribution of a random variable X is given as
`{:(X,-5,-4,-3,-2,-1,0,1,2,3,4,5),(P(X),p,2p,3p,4p,5p,7p,8p,9p,10p,11p,12p):}`
Then, the value of p is

Options

A. `1/72`
B. `3/73`
C. `5/72`
D. `1/74`

Answer

To find the value of \( p \) in the given probability distribution of the random variable \( X \), we will follow these steps:

### Step 1: Write down the probabilities and their corresponding values
The random variable \( X \) takes on the values \(-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5\) with corresponding probabilities:
- \( P(X = -5) = p \)
- \( P(X = -4) = 2p \)
- \( P(X = -3) = 3p \)
- \( P(X = -2) = 4p \)
- \( P(X = -1) = 5p \)
- \( P(X = 0) = 7p \)
- \( P(X = 1) = 8p \)
- \( P(X = 2) = 9p \)
- \( P(X = 3) = 10p \)
- \( P(X = 4) = 11p \)
- \( P(X = 5) = 12p \)

### Step 2: Set up the equation for the total probability
According to the properties of probability distributions, the sum of all probabilities must equal 1. Therefore, we can write:
\[ p + 2p + 3p + 4p + 5p + 7p + 8p + 9p + 10p + 11p + 12p = 1 \]

### Step 3: Combine like terms
Now, we combine all the terms on the left side:
\[ (1 + 2 + 3 + 4 + 5 + 7 + 8 + 9 + 10 + 11 + 12)p = 1 \]
Calculating the sum of the coefficients:
\[ 1 + 2 + 3 + 4 + 5 + 7 + 8 + 9 + 10 + 11 + 12 = 66 \]
Thus, we have:
\[ 66p = 1 \]

### Step 4: Solve for \( p \)
To find \( p \), we divide both sides of the equation by 66:
\[ p = \frac{1}{66} \]

### Conclusion
The value of \( p \) is:
\[ \boxed{\frac{1}{66}} \]