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 follow these steps:

### Step 1: Write down the probability distribution
The random variable \( X \) takes the values:
\[ X = \{-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5\} \]
with corresponding probabilities:
\[ P(X) = \{p, 2p, 3p, 4p, 5p, 7p, 8p, 9p, 10p, 11p, 12p\} \]

### Step 2: Set up the equation for total probability
The sum of all probabilities must equal 1:
\[ p + 2p + 3p + 4p + 5p + 7p + 8p + 9p + 10p + 11p + 12p = 1 \]

### Step 3: Combine like terms
Add all the coefficients of \( p \):
\[ (1 + 2 + 3 + 4 + 5 + 7 + 8 + 9 + 10 + 11 + 12)p = 1 \]
Calculating the sum of the coefficients:
\[ 1 + 2 + 3 + 4 + 5 + 7 + 8 + 9 + 10 + 11 + 12 = 72 \]
Thus, we have:
\[ 72p = 1 \]

### Step 4: Solve for \( p \)
To find \( p \), divide both sides by 72:
\[ p = \frac{1}{72} \]

### Final Answer
The value of \( p \) is:
\[ \boxed{\frac{1}{72}} \]